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CRYSTALS AND THE FINE-STRUCTURE 
OF MATTER 


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CRYSTALS AND THE 
FINE-STRUCTURE 
OF MATTER 


FRIEDRICH RINNE 


PROFESSOR OF MINERALOGY IN THE UNIVERSITY OF LEIPSIC 


TRANSLATED BY 


WALTER 5S. STILES 


WITH A DRAWING BY A, DURER, AND PORTRAITS OF THE LEADING INVESTIGATORS 
IN THE STUDY OF FINE-STRUCTURE AND 202 FIGURES 


NEW YORK 
E. P. DUTTON AND COMPANY 
PUBLISHERS 


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PREFACE 


to the scientific publica yearago. The book 

set out to consider experiments on the fine- 
structure of matter from a new and essentially 
crystallographic standpoint, corresponding to its 
title ‘‘Crystals as types of the fine-structure of 
matter.”’ 

The friendly communications I have received, the 
reviews in the press, and the fact that translations 
of the book have been prepared, will testify to the 
good reception given to the work by the public. 
The demand for the first edition being so great, I 
have thought it well to extend the work consider- 
ably for the new issue. 

Historical details are introduced, the treatment 
of crystallography is amplified, tabular summaries 
and sections on the atom domain and _ stereo- 
chemical axes are added. Instructive cases of poly- 
morphism are also described. In addition, the book 
contains many new explanatory figures, and portraits 
of some leading scientists. By these changes the 
work is not, I think, increased in size by more than 
half of its previous bulk, without at the same time 
being enriched by many glimpses into the interesting 
relations of the science of fine-structure. 

The need for a treatment of the subject easily 


Vv 


a oer first edition of this work was submitted 


67055 


"i CRYSTALS AND MATTER | 


understood by the general reader is, however, con- 
stantly observed. For those more closely interested 
a list of text-books on crystals has been included. 
An index is appended. 

I hope that this new and much enlarged edition, 
the preparation of which has been a labour of love 
for me, will help, as did its predecessor, both by its 
text and diagrams, in placing in the right light the 
physical, or, rather, the natural philosophical side of 
crystallographic science, and in acquiring new friends | 
for the very fertile study of the fine-structure of 
matter. 

To these preliminary remarks I add my thanks 
for the advice offered to me by the reviewers, whose 
suggestions have been willingly followed, together 
with the expression of my gratitude to my assistants, 
Dr. K. H. Scheumann and Dr. E. Schiebold, for 
friendly help in the editing of this new edition. The 
Saxon Academy of Sciences facilitated the publica- 
tion of the book by the loan of a large number of 
printing blocks. 


INSTITUTE OF MINERALOGY AND PETROGRAPHY 
IN THE UNIVERSITY OF LEIPSIC 
IN THE SPRING OF 1922 


NOTE 


THE translator wishes to tender his best thanks to 
Professor Rinne for very kindly reading through the 
proofs, and offering many helpful suggestions. Most 
of these have been adopted and have materially 
assisted in the work of translation. 


W. OAS bes 


CONTENTS 


CHAP. PAGE 
PREFACE . ; : : ; é ; : : Vv 

I. INTRODUCTION . : . ; ; : I 
II. THE STUDY OF FINE-STRUCTURE (LEPTOLOGY) . : 5 
III. CRYSTALLOGRAPHY AND LEPTOLOGY 6 


t, Historical,.p.6. 2. The Laue Effect, PA LD ise 
M. v. Laue, p. 13. 4. Further X-ray methods, 
p.16. 5. X-ray results in the domain of chemis- 
try, p. 20. 6. Crystals as stereochemical types, 
p. 25. 7. Outlines of general crystallographic 
morphology, p. 28. 8. Extension of the mani- 
fold of crystallographic types by twin formation, 
Pp. 35: 

IV. FINE-STRUCTURAL UNITY OF MATTER ‘ vty) es (8) 


1. Fine-structure of amorphous bodies Ged 
with that of crystals, p. 39. 2. Physical investi- 
gations on the general forms of atoms and mole- 
cules, p. 43. 3. Atom domains, p. 46. 4. Differ- 
ence between the structure of individual leptons 
and crystals, p. 49. 


V. THE GENERAL CHARACTERISTICS OF THE FINE-STRUCTURE 
OF MATTER ; . ‘ ; : ; ad eo 

y. Crystals, p. 54. 2. Gases and liquids, p. 59. 

3. General character of the fine-structure, p. 60. 
VI. THE SERIES OF TRANSFORMATIONS OF MATTER . 61 

1. States of aggregation ; gases, liquids and Henna 

crystals, crystals, p. 61. 2. Discontinuities of 

lower order in theseries. Polymorphism. Enan- 

tiomorphy, p. 69. 3. Review and unified con- 

ception, p. 76. 

VII. GENERAL TECTONIC ARRANGEMENT IN THE FINE-STRUC- 
TURE OF CRYSTALS : 78 

1. Analysis of een emia! Ties p. 8, 

2. Valency, p. 80. 3. Molecules in the crystal. 

Structural groups (leptyles}. Lattice types, 


p. 87. 
VIII. ASSOCIATION OF THE FINE-STRUCTURAL PARTICLES IN 
MIXED CRYSTALS AND OUTGROWTHS ON CRYSTALS . 97 


1. Transitions between chemical combination and 
physical mixture. Mixed crystals, p. 97. 2. 
Physico-chemical significance of the mixed 
crystal, p. 98. 3. Out-growths with substances 
not isomorphous, p. 102. 

b vii 


Vill 


CHAP. 


IX. 


Xi, 


XII. 


>, bee 


XIV. 


XV. 


CRYSTALS AND MATTER 


MORPHOTROPY . : : : : , : 
1. Historical, p. 105. 2. Stereochemical axes, 
p. 106. 3.:Morphotropic constructions, p. 107. 

4. Examples from the mineral world, p. 111. 


. ISOTYPY 


1. Crystal Pans Eyota pastes rae Sryeees foun 
as stability types, p. 115. 2. Fine-structure 
groups as stability types, p. 116. 

CRYSTAL GROWTH AND SOLUTION : ; ; 

1, Pure crystal growth, p. 121. 2. Mixed crystal 
growth, p. 128. 3. Collective crystallisation, 
p.131. 4. Crystal solution, p.133. 5.Summary, 
Pp. 137. 

CHEMICAL ACTIONS ON CRYSTALS 

1. Anisotropy of chemical actions on crystals, D- 139. 
2. Anisotropic chemical reactions of molecules, 
p-140. 3. Structural rigidity of electrons, atoms, 
molecules, and crystals, p. 141. 4. Crystallo- 
graphic chemical changes. Undermining and 
reconstruction, p. 143. 5. Resistance to mechani- 
cal disruption and chemical attack, p. 154. 


AN ATTEMPT TO FORM SOME IDEA OF THE COURSE OF 
CHEMICAL REACTIONS FROM OBSERVATIONS ON 
CRYSTALS 


1. Chemical SreMeay notions! Dp. 158. | 2. 
chemical processes and discontinuous reactions. 
Mass action and catalysts. Heat as a catalyst, 
p.160. 3. Crystallographic indicators of chemical 
processes, p. 163. 


ANALOGY OF THE MORPHOLOGICAL ACTION OF PHYSICAL 
AND CHEMICAL FIELDS ON CRYSTALS 


1. Thermal influences on the crystal form, p. 170. 
2. Chemical influences on the crystal form, p. 174. 
3. Comparison of the thermal and chemical in- 
fluences on the crystal form, p. 176. 


CRYSTAL PHYSIOLOGY AND THE CLASSIFICATION OF 
ATOMS. 


1. einen uel gat Uheaioieey of the cndat cated p. 
180. Electrons, atoms, and molecules, p. 181. 
3: Kiam types, p. 182. 4. Atom sub-types, 
p.185. 5. Elements, p.185. 6. Normal mixture 
and separation of isotopes, p. 186. 


CONCLUSION 


INDEX 


PAGE 
105 


II5 


I21I 


139 


158 


170 


180 


187 
IQI 


LIST OF PLATES 


Dr. W. C. RONTGEN . 


MELANCHOLY 
From an Ferrero b by Apert Diirer. 


RENE Just Haty 
P. v. GROTH 

A. SCHOENFLIES 
E, v. FEDOROW 
M. v. LAUE 

FIGS. 15 AND 16 
P. DEBYE 

P. SCHERRER 


Sirk WILLIAM HENRY BRAGG 
From a Photograph by Elliott & Fry, Ltd. 


WILLIAM LAWRENCE BRAGG 
From a Photograph by Elliott & Fry, Ltd. 


G. v. TSCHERMAK 

Fias, 50, 51 AND 52. 
Fic, 86 

Fics. 163 AND 164 
Figs, 165, 166 AND 167 


Frontispiece 


TO FACE PAGE 
I 


22 


26 
38 
66 
140 


144 


= OF Te 
we “Ualvexaite uF wis 
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MELANCHOLY 


FROM AN ENGRAVING BY ALBERT DURER 


CRYSTALS AND THE 
FINE-STRUCTURE OF MATTER 


I. INTRODUCTION 


HE reader sees before him a reproduction of a 
drawing by the great German master, Albert 
Diirer, whose art has here represented the 
problems of natural science and technical practice, 
together with the simple means available in his time 
for their elucidation. Among the symbols in the 
picture we observe in a very prominent place a 
gigantic crystal, which is surely an indication that 
Diirer saw a scientific problem of the first importance 
in the solution of the riddle of this regular form. 

Over the whole scene rests a gloomy air of 
speculative resignation, the Faust-like expression of 
the feeling that we, ultimately and in spite of fervent 
endeavour, “nothing can know.” The artist has 
called his picture “‘ Melancholy.” 

A Durer of our own day would certainly have 
drawn more hopefully. It would seem as though the 
darkness enveloping the great mysteries of nature 
were lifting a little. By studying crystals and using 
them in other investigations much light is thrown on 
the ultimate significance of matter and on the nature 
of the forces which, acting from one minute particle 


to another, result in the coherence of the universe. 
of . 


2 CRYSTALS AND MATTER 


Crystals are proving themselves more and more 
the ideal substances of physics and chemistry. Even 
before the last great discoveries in this branch of 
knowledge the late Woldemar Voigt, Professor of 
Theoretical Physics at G6ttingen, who had a unique 
knowledge of this subject, drew attention to the 
exceptional regularity of the crystalline parts of 
matter in a fine simile, which is quoted below : 

‘Imagine a couple of hundred picked violin 
players in a large room, all playing the same piece 
of music on faultless instruments, but beginning 
simultaneously at widely different places, and start- 
ing afresh each time they reach the end. The finest 
ear would be unable to recognise the piece actually 
played in this uniform medley of sound. Now such 
music is presented to us by the molecules of gaseous, 
liquid, and ordinary solid bodies. A crystal, on the 
other hand, corresponds to the orchestra described 
above when it is guided as a whole by one able 
conductor, so that all eyes hang on his slightest 
gesture, and all hands play the same bar. In this 
way the melody and rhythm of the piece presented 
become completely effective, the number of the per- 
formers not hindering but intensifying the result.”’ 

This picture makes it clear how crystals can 
present a large series of phenomena which are absent 
in other bodies, and that, in them, certain character- 
istics are developed in wonderful variety and elegance, 
which elsewhere occur only as monotonous mean 
values. This fact is briefly referred to in Voigt’s 
text-book at the end of the chapter on the esthetic 
side of crystal science as follows :— 

‘In my opinion the music of physical laws in no 
other branch achieves harmonies so full and rich as in 


INTRODUCTION 3 


crystal physics.’ The full truth of his view has been 
strikingly proved in the last ten years; it could not 
have been brought to notice in more impressive 
fashion than by the results of the researches initiated 
by M. v. Laue on the action of X-ray impulses in 
crystals. The fine-structure of crystals acts like 
a grating diffracting the X-rays. In this “ Laue 
effect,’’ which will be discussed later, crystalline sub- 
stances are seen to be the best ordered materials. 
It is easily understood that, since these experimental 
demonstrations of the constitution of crystals, the 
investigations not only of mineralogists but also of 
physicists and chemists have been especially con- 
centrated on the crystal form. Further magnificent 
results have been obtained which are extensions of 
the discovery of M. v. Laue and his fellow-workers, 
W. Friedrich and P. Knipping. These matters cer- 
tainly merit the widest dissemination in scientific 
circles. Many expositions with this object have 
already appeared. Besides taking a share in the 
work of research, I have frequently done my best 
to promote the’ same end by means of notes in 
journals and papers. 

The present small work has, in addition, a 
wider purpose. In a treatment which aims at 
being comprehensible, as far as possible, to general 
readers, while at the same time offering much new 
material to fellow-students, the attempt is also made 
to deduce from the properties of crystals the main 
characteristics of the fine-structural constitution of 
matter. 

With this in view the title has been slightly altered 
from that of the first edition. The aim of this book 
is thus to show that crystals exemplify not only the 


4 CRYSTALS AND MATTER 


morphological but also the physical and chemical 
constitution of matter. The crystal is therefore 
treated from the standpoint of its fine-structure, 
and a discussion of amorphous bodies (for example 
gases and liquids) is included. 


II. THE STUDY OF FINE-STRUCTURE 
(LEPTOLOGY) 


study, or leptology, because the customary term 

stereochemistry does not cover completely the 
field of the inquiry intended. In stereochemistry 
we study the form and arrangement of the particles 
comprising various substances in order to explain 
thereby their chemical properties. Alongside this 
science, another has arisen, which may justly be 
termed stereophysics, and this also is concerned with 
the constitution and association of the particles of 
matter, dealing especially with their movements and 
physical properties (e.g. crystal optics). Finally, a 
third allied subject is included, namely, crystal 
structure, or the study of the fine-structure of 
crystalline bodies from the point of view of their 
geometrical relations. 

Thus from the trunk of Greek Atomic Theory, 
which is more than two thousand years old, three 
branches of knowledge have sprung forth and blos- 
somed—stereochemistry, stereophysics, and crystal 
structure. Their intimate relations to one another 
justify a name to include them all. If now the 
particles of matter, the electrons, atoms, ions, and 
molecules which constitute gases, liquids, and crys- 
tals are termed collectively fine-structure-particles or 
leptons (from Xemros, fine, delicate), the name sug- 
gested above, fine-structure-study or leptology indi- 
cates’ precisely the end in view. 

5 


I INTRODUCE the expression fine-structure- 


Ill. CRYSTALLOGRAPHY AND LEPIOUCOGGS 


HISTORICAL 


EARLY two thousand five hundred years 
Nig. in ancient Greece, where Babylonian, 

Persian, and Egyptian wisdom was _ asso- 
ciated with the Greek genius, philosophical theories 
as to the fine-structure of matter were extant. 
These theories were due to Democritus and to his 
friend Leucippus, and also to Epicurus, who lived 
some hundred years later. Their writings have not 
come down to us; their ideas, however, are pre- 
served in the poem “ De rerum natura ’’ of Lucretius 
(96-55 B.C.). 

According to this they assumed all matter, 
considered in its finest state, to have a granular 
structure, and to consist invariably of aggregates 
containing large numbers of atoms. These very 
small particles are, in their opinion, extended but 
indivisible, unchangeable, and for different sub- 
stances, of different shape, magnitude, and weight. 
They oscillate and move about at random. Between 
them is empty space. 

Under the influence of Aristotle this theory 
retreated into the background, and it found no 
favour in intellectual circles of the Middle Ages 
until scientists, such as P. Gassendi (1592-1655), 
and the great physicist, I. Newton (1642-1726), 
endowed the old fundamental idea with a new 


validity. As a consequence, closer relations between 
6 


CRYSTALLOGRAPHY, LEPTOLOGY 7 


leptology and crystallography were soon estab- 


lished. 


In this connection Ch. Huygens (1629-1695) 


was able to explain the form of calc-spar, its cleavage, 
and the variation of its hardness and double refraction 
with direction, by assuming a regular packing of 
(a conception which 


spheroidal particles 
Barlow and _ Pope 
also employed suc- 
cessfully). Later, 


the principal founder 
of scientific crystallo- 
graphy, Abbé René 
Just Haiiy 


(1743- 


Fic. 1.—Fine-structure of calc-spar 
according to Ch. Huygens. 


later 


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(usannme 
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Fic, 2. —Crystalline fine-structure according to 
Re ye tiauy: 


1822,) endeavoured to establish his expositions on 


a fine-structural basis. 


+) 


“additive molecules 


He thought of crystals as 
constructed of, for the most 


part, contiguous building stones, the form of which 
determined the cleavage, and he deduced by means 
of “ decrescence”’ (i.e. regular, step-like reduction in 
the size of molecular plates on a nucleus) the crystal- 


line forms from such ‘ 


ordinary fineness of 


‘Primitive Bodies.’’ 


The extra- 
the staircase structure renders 


the boundary planes perfectly smooth to the eye, 
and even to the most sensitive optical tests. 


8 CRYSTALS AND MATTER 


An experimental law of crystallography, that 
of the “‘ simple rational axial sections of the crystal 
faces,’ 1 found in the above its obvious explanation. 
We understand by this law that in the external form 


coc eoc 
Fic. 93, Fic. 4. 
Illustration of the law of simple rational axial sections of crystal faces. 


of crystals, surfaces arbitrarily placed do not occur, 
but only those which stand in a definite relation to 
one another. In contrast to the freedom of an archi- 
tect who, in building a house with the roof D, as 


Fic. 5.—The relationship of the faces of an aragonite crystal on three inter- 
secting axes. 


shown in Fig. 3, can at will give a small or large 
value to its angle of inclination, in crystal structure 
(Fig. 4) only certain slopes are possible, such as those 


1Commonly known as the “ Law of Rational Indices.” 


RENE JUST HAUY 
Born 28th February, 1743, Died 3rd June, 1822 


sé 


,. , a H ‘ 


a 


Wisivchou: oe acumen - 


4 » ae a 


CRYSTALLOGRAPHY, LEPTOLOGY 9 


from 6 to c, from 6 to $c, 2c, 2c, cc, or, in other words, 
according to simple rational ratios. Similar remarks 
hold for the other external faces of the crystal with 
respect to the points of section on the spacial axes 
a, b,c. If the units of these are marked out, say, 
for aragonite (Fig. 5), by means of the axial points of 
the face O, = a: 6:c in which ratio the length of 6 
is set equal to unity, and each of the other surfaces 
be imagined displaced parallel to itself till it passes 
through the end point 10, then experimentally all 
such surfaces cut the axes a and 
cin the manner discussed above. 
For example, the face 6 cuts 
these axes In a:b: oc; m,, 
Meer COC Dy, 1M cod. UC. 
Sritieed..0 2 2c; and'so.on: pas 

A consideration of Fig. 6 a@étetetdteretseey2 7 
renders this law of crystallo- Crys Bas 
graphy, which restricts in a ee nat RAR TEL 
far-reaching fashion the exter- 
nal form of crystals, immediately intelligible accord- 
ing to the ideas of R. J. Haity. 

Against the great advantage of the agreement 
between the Hatiy theory and crystallographic prac- 
tice several considerations, especially those of a 
physical nature, must be advanced. The compres- 
sibility of the crystals, and their volume changes, 
with change of temperature, make their structure 
according to the Hatiy scheme improbable. How- 
ever, these difficulties could easily be got over, as the 
founder of the theory himself pointed out, if the 
closely-packed ‘‘ additive molecules ”’ are replaced at 
their centres by particles freely oscillating, which 
mutually maintain each other in this arrangement. 


10 CRYSTALS AND MATTER 


Thus we arrive at the idea of space-lattices as systems 
of particles, arranged periodically in three dimensions 
(Fig. 7). oe 
Such a system is formed by the points of inter-- 
section of three families of planes, each family 


Fic. Uae HOPE En) Fic, 8.—Molecular lattice according 
to A. Bravais. 
consisting of parallel planes, the distance between 
consecutive members being constant. According to 
A. Bravais (1811-1863), the space-lattice particles 
consist of chemical molecules, the symmetry of which 
is determined, by) ices 
stallographic considerations 
(Fig. 8). The Munich phy- 
sicist, L. Sohncke (1842- 
1897), and his mineralogical 
colleague, P. v. Groth, on 
the other hand, replace the 
Bravais molecular lattice by 
what they call point sys- 
tems, which are atom lat- 
ie scree em seeorling tices placed regularly one 
inside the other, as indicated 
in Fig. 9, where the scheme includes only two types 
of atom. 
At the same time, the necessary geometrical 


CRYSTALLOGRAPHY, LEPTOLOGY 11 


representation of all space groupings possible in 
accordance with crystallographic laws was carried 
out. A. Schdénflies? has given in a classical treatise 
the whole system of space-lattice arrangements. The 
same work has also been accomplished by the Russian 
investigator, E. v. Fedorow. 


ee CP ECL: 


The year 1912 came as the great harvest year in 
the physics of space-lattice ideas. This led to the 
ever-memorable researches instituted by M. v. Laue, 
and carried out jointly with P. Knipping and 
W. Friedrich, in Munich, on the use of crystals as 
diffraction gratings for X-rays. A polychromatic 
impulse of the radiation is split up by diffraction 
at the particles of 
the crystal grating 
into a spectrum. of 
monochromatic rays. 
These, received on a 
photographic plate, 
give rise after de- 
velopment to a figure 
symbolic of the Fic. 10.—The ‘“‘ Laue Effect.”’ 
atomic arrangement, 
in the form of a Laue diagram (Fig. 10). This diffrac- 
tion effect can be considered ? formally as a reflexion 


? 


1A. Schénflies, “‘ Kristallsysteme and Kristallstruistur,”’ 1891. 

2 Thus each point in a Laue diagram is to be regarded as the 
point of impact of X-rays which have been reflected at a structure 
plane in the crystal. Fig. 12 makes this clear for two planes, E,F, 
and E,E,. The primary ray P is reflected at E,E, to R,, and at 
E,E, to R,. R, and R, are thus the points of impact of secondary 
rays on the photographic plate. Rhythmic arrangements of struc- 
ture planes are emphasised in the radiogram by corresponding 


12 CRYSTALS AND MATTER 


of the incident rays at the planes of atoms in the 
crystal, in which case, however, a reflected ray is 
obtained only if the condition n\ = 27 sin a is 
fulfilled? (Fig. 11). This aspect was especially em- 
phasised by W. H. and 
W. L. Bragg. 


P 


Fic, 11.—Reflexion of X-rays. Fic. 12.—Diagram showing re- 
flexion of an X-ray at two struc- 
tural planes of a crystal. 


In this way the space-lattice idea of crystallo- 
graphy became the starting-point of an extraordinary 
development of physical science ; for not only the 
nature of X-rays as wave phenomena, but also the 
actuality of the atom was experimentally proved, 


repetitions of reflected rays of the same intensity ; the Laue diagram 
of beryl (Fig. 14) is an especially beautiful example of this. 

The customary arrangement of the points on elliptical curves is 
easily explained with Fig. 13. In this S,s, is the primary ray, which 
is reflected along Ks, by a plane cutting the plane of the paper 
at right angles and in the line Kz. If this plane be rotated about 
Kz, the reflected ray traces out a conical surface with Kz as axis. 
The receiving photographic plate PP cuts this cone in an ellipse 
S,\S,. For other inclinations of the line Kz, parabolas, hyperbolas, 
or straight lines are obtained for the series of reflexions from those 
planes which run parallel to some one direction (corresponding to 
Kz) ; or which, as the crystallographer says, lie in a “‘ zone.”’ 

1n=I1, 2, 3...2XA= wave lengths. v» = distance between 
reflecting planes. a = glancing angle. ‘The path difference for the 
two rays I and 2 is obviously equal to wu, thus it equals 2y sin a 
(Fig. 11). 


E. v. FEDOROW 


me La f 


oun THE. LORNA 
= JF HE 
| -HakeaL Lay Of i - ‘ais 


CRYSTALLOGRAPHY, LEPTOLOGY 13 


once and for all, by the Laue effect. The existence 
of atoms is as certain now as that of the macrocosm 
of the starry heavens. Laue’s experiment may with 


Fic. 13.—Formation of zone curves in Laue diagram. 


justice be described as a solemn deposition of nature 
concerning its most intimate structure. 


M. v. LAUE 


Under these circumstances it will be of interest 
to the reader to hear something more of the young 
German scientist and his celebrated researches, which 
opened up an unlimited field for investigation in 
the fine-structure of matter by the application of 
X-rays. 

M. v. Laue was born on the gth October, 1879, 
at Pfaffendorf, near Coblenz. While a student in 
Berlin he received the greatest encouragement in his 
scientific work from the creator of the quantum 
theory, M. Planck, whose assistant he was from 


14 CRYSTALS AND MATTER 


1906 to 1909. When v. Laue removed to Munich in 
1909, stimulated by the work of Réntgen and the keen 


Fic. 14.—Laue diagram of the end surface of a crystal or beryl. Photographed 
by F. Rinne. 

interest of Sommerfeld in X- and y-rays, his attention 

was directed to the question of the nature of these 

rays. Moreover, it chanced that at Munich Uni-— 


surface 


y =e ’ : p 
i a i 1 j , : : ; i i 
{ -* i is 
7 {ow ; — rh { a boii} *y oe 
; = 2 % > . 7 
a 7 a ar 
- i by fl 
pow” 
’ “7 ; 7 _ 
“ 
' é ~ ba 
: ‘ ++ F : 
+ > 
: ‘ J st 
’ é i 
4 \ | | 
/ é' : 
» Oty if 
; i +5 fi 
‘ 
J we - ‘ 
j 
1 
if _ - . 
+ 
x 
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es my 4 
et 
i 
i 
Z Py , 
$ F , 
i { 
f i - 
i } t 
oe THE © oa le 
leu 
 GaivewalY id WLLRUIS LP = 
ol 
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t Hk 
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5 ‘ 
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7 nes Y - F 
mye ae. . 
4 
4 ~ 
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® 
: 4 4 > CL 
aD en aes i 
: e ® 7 ' 
y 7 +m se - % e 
ho ., 2 oI i - 
s 7 Pg - wu ® 
. . Fl : } - wi |b - 7 
A i >, é =i es el wwe 
aT the ae ins he i. : eee 
7 - ' : x ae, rye wie a e -_ 
! = =. fa ; 
Ps : : = 4 a i k : eal 3 


. 
ie? 
q * 3 a - if - 


7 3 
FT 7 ) : 7 4 ® ie ¢ ' ~~ 
» ‘ P " 


CRYSTALLOGRAPHY, LEPTOLOGY 15 


versity, P. v. Groth, in his lectures and writings, was 
emphasising the space-lattice idea, and this ultimately 
took its place in the circle of ideas of the physicists 
there. In this way it happened that, when v. Laue 
was visited in February, 1912, by his colleague, 
P. P. Ewald, they discussed the studies of the latter 
as to the relations of long electro-magnetic waves 
in a space-lattice, and the question occurred as to 
the action of such waves, which are short compared 
with the dimensions of the lattice. Laue’s optical 
knowledge told him that grating spectra must arise ; 
he expected interference phenomena in the passage of 
X-rays through crystals, and mentioned this to Ewald. 
Copper vitriol served as the material for the first 
experiment, large regular pieces of this being easily 
obtained. Friedrich and Knipping left the direction 
of the incident beam to chance. On the photographic 
plate behind the crystal there appeared in the first 
experiment the expected grating spectra. In the 
continuation of the work, Friedrich and Knipping 
investigated - regular crystals having the highest 
possible symmetry, such as zinc-blende, and illu- 
minated them in the direction of the crystallographic 
axes, with the beautiful results submitted to the 
reader in the original figures 15 and 16. 

By applying here the results for the ordinary and 
the crossed grating the theory of the phenomena was 
immediately obtained, and Professor Sommerfeld was 
able, on the 8th June, 1912, to lay before the Munich 
Academy the joint work of Friedrich, Knipping, and 
v. Laue on the interference of X-rays. 

In 1912 M. v. Laue obtained the Professorship 
of Theoretical Physics at Zurich University. From 
1914 to 1919 he acted as Professor in the same 


16 CRYSTALS AND MATTER 


subject at Frankfurt University ; a summons to 
Berlin University brought him to his present position. 
A further recognition was accorded him in the Nobel 
Prize for physics for the year IgI4. 


FURTHER METHODS IN X-RAY WORK ON CRYSTALS 


It is to a number of investigators, happily an in- 
creasing number, that we owe extensions of the method 
of the fundamental research of Laue, Friedrich, and 
Knipping. W. H. and W. L. Bragg‘ used plates cut © 


Fic. 17.—Rotation spectrogram of adularia. After E. Schiebold. 


in known directions from crystals, and rotated then 
on a spectrometer about an axis lying in the surface 
of the specimen. As soon as the angle between the 
plane of the specimen and one of the incident X-rays 
satisfied the equation m\ = 27 sin a reflexion occurred. 
The reflected ray lying in the plane of incidence was 
detected by means of a cylindrical ionisation chamber, 
and its inclination determined. The quantity of 


1W. H. and W. L. Bragg, “ X-Rays and Crystal Structure.” 


CRYSTALLOGRAPHY, LEPTOLOGY 17 


ionisation served aS a measure of the intensity of the 
reflected radiation. 

Other rotation methods with photographic deter- 
mination of the direction of the reflected rays have 
been elaborated, especially by H. Seeman and E. 
Schiebold. 

Interesting researches on substances possessing a 
fibrous or flaky structure, such as occur in nature, 
in plants, and animals, or can be prepared by drawing 
or pressing metals, have been carried out by members 
of the Research Institute in the Chemistry of Fibrines 
at Dahlem, near Berlin; among these, Polanyi and 
his collaborators, Becker, Herzog, and Jancke, may 
be specially mentioned.! 


1 Substances built up of fibres or flakes, which lie with all their 
fibre axes or flake normals parallel, but otherwise indifferently, give 
on the passage of X-rays perpendicular to this direction an X-ray 
effect similar to that of a crystal plate which is rotated about an 
axis passing through it. The series of reflexions, one after the other, 
produced by the latter, as a result of the rotation, are shown by the 
pack of fibres or flakes simultaneously. All fibres or flakes the struc- 
ture planes of which satisfy the equation n\ = 2y sin a give rise to 
reflected rays. As was shown by H. Seeman and E. Schiebold, in 
particular for rotating plates, and by M. Polanyi for fibrous sub- 
stances and stretched metals, we get in this way characteristic 
diagrams. The reader more closely interested is referred to Fig. 18. 
In this PSt denotes the primary ray. It is reflected at surfaces 
whose normals N,, N,, etc., corresponding to various positions of 
rotation about the rotation axis DA (or the axis of the fibres, as 
the case may be), are shown in the figure. The incident ray, the 
normal to the surface, and the reflected ray, lie in every case in one 
plane, e.g. the ray S,, corresponding to N3 liesin the plane E,. If 
monochromatic radiation be used and only reflexions of a given order 
be considered (i.e. if we assume definite values for \ and ” in m\ = 2r 
sin a), then reflected rays arise which pass through the intersections 
of the a-circle with the great circles corresponding to various planes 
of incidence. Take, for example, N = 3. In this case the inter- 
sections on the circle E, are, on the sphere, S,S’; ; on the photographic 
plate PhPl, S,S’,. For other values of ”, and therefore of a, other 

2 


18 CRYSTALS AND MATTER 


Of the greatest importance is a method emanating 
from P. Debye and P. Scherrer, in Géttingen, which 


‘ 


, ‘ 
‘ 
i) 


Fic, 18.—Explanation of the rotation spectrogram. After E. Schiebold, 


Fic. 19.—Reproduction of Fic. 20.—Reflexion cone of X-rays obtained 
a Polanyi diagram. with crystal powder. 


renders the investigator independent of the possession 


of oriented plates or of crystals with any regularity 
at all. 


reflexions will arise. Taking all the intersections into consideration, 
they lie on the curve similar to a lemniscate, shown in the figure. 
In general, of course, there are four reflected rays; in special 
positions of the reflecting planes, only two. Compare S, and S’,, S; 
and S’,, in Fig. 18, also Fig. 19. 


CRYSTALLOGRAPHY, LEPTOLOGY 19 


These scientists employed powders of the finest 
crystalline particles, such as can be obtained by 
precipitation from solution or by continued pul- 
verisation. In such a case the structure planes re- 
flecting the X-rays lie at random in all directions. 
Those of them, however, which are inclined to the 
primary rays at the glancing angle a, satisfying the 
equation m\ = 2rvsina, give rise to a reflected ray ; 
and since such positions occur all round the primary 
ray, a cone of rays is produced, as P. Debye and P. 
Scherrer showed, instead of the | 
single reflected ray obtained in 
the Laue diagram. This effect is 
that which would be obtained if 


Fic. 21.—Camera for the Debye- Fic, 22.—Unrolled film with Debye- 
Scherrer method. Scherrer diagram. 

a Laue diagram were rotated about its central normal. 

Since monochromatic light is employed, the hollow 

cones of reflected rays obtained are thin, and cut 

the interposed photographic plate in separate circles 

(Fig. 20). 

In order to make the distance travelled by all the 
reflected rays to the receiving surface the same, and 
to facilitate photographic registration, a cylindrical 
camera, with the specimen as a small roll at its 
centre, is employed, following the suggestion of 
P. Debye and P. Scherrer. A film is placed round 
the inner wall of the camera; after exposure and 
photographic development this is unrolled, examined, 
and measured (Figs. 21 and 22). Independently of 
Debye and Scherrer, the American investigator Hull, 


20 CRYSTALS AND MATTER 


has devised a similar method. W. H. Bragg has also 
combined the Debye-Scherrer with his spectrometric 
ionisation method. , 

With respect to the manipulation of the experi- 
mental results, the reader interested further is 
referred to the more detailed works on the subject. 


Phot +late (elevation) 


GRC AD 


topor N20) 
1 
| | Phot. 4 late (see tun 


A Phot. sil (cylinder) ——~ 
ee 
a 


Incident 
Leam 


Fic. 23.—Construction of the Debye-Scherrer diagram of powdered 
graphite. 


X-RAY RESULTS IN THE DOMAIN OF CHEMISTRY 


The X-ray spectra of H. Moseley (1888-1915) 
provided chemistry with the natural series of the 


1W. H. and W. L. Bragg, “‘ X-Rays and Crystal Structure.” 
E. Marx, ‘‘ Handbuch der Radiologie, Band 5 (Kathodenstrahlen und 
Rontgenstrahlen).’”’ P. Niggli, ‘‘Geometrische Kristallographie des 
Diskontinuums.”’ F. Rinne, “ Einfuhrung in die Kristallographische 
Formenlehre sowie Anleitung zu Kristallographisch-optischen und 
roéntgenographischen Untersuchungen,”’ 5th edition, published by 
Dr. M. Jaenecke, Leipsic. Works by P. P. Ewald and E. Schiebold 
on X-rays and Crystals are in course of preparation. 


DR? PP. DEBYE 


Professor of Physics in the University of 
Zurich 


DR. P. SCHERRER 


Professor of Physics in the University of 
Zurich 


THE LIBRARY 
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CRYSTALLOGRAPHY, LEPTOLOGY 21 


¢ 


elements according to their ‘‘ atomic numbers,” an 
arrangement to which there are no exceptions up to 
the present time. In L. Meyer’s and D. J. Mendeléeff’s 
“Natural System of the Elements,’ where they are 
set down in the order of increasing atomic weight, 
the arrangement being broken at certain points and 
so split up into series, there were several very awk- 
ward anomalies. Contrary to the guiding principle 
of the arrangement according to increasing weight, 
the properties of argon necessitated its being placed 
before potassium in spite of its higher atomic weight. 
For similar reasons cobalt came before nickel, and 
tellurium before iodine. This anomaly has now been 
removed. The spectrometric diffraction of X-rays, 
using crystal plates as gratings, with anticathodes of 
the various substances, proved that the elements are 
ordered by their X-ray spectra in complete accord- 
ance with the corrected Meyer-Mendeléeff table. The 
square root of the frequency of the spectral lines is 
a linear function of the atomic number in the sys- 
tem. Accurate determinations in particular by the 
Lund physicist, Manne Siegbahn, have definitely fixed 
this extremely important scientific result, “the 
atomic number,’’ which constantly characterises the 
elements. In addition, these X-ray investigations. 
rendered possible determinations of exact stereo- 
chemical formule by a physical method. W. H. and 
W. L. Bragg have led the way in this with remarkable 
ingenuity. They measured the 7 values of crystals, 
for example, of fluor-spar, for various directions, 
such as perpendicular to the cubic, rhombododeca- 
hedral and octahedral surfaces, and were able to 
conclude therefrom, with the help of the intensity 
relations of the appropriate spectra, the positions of 


22 CRYSTALS AND MATTER > 


the atoms. Later the fine-structural constitutions of 
rock-salt, fluor-spar, zinc-blende, diamond, calc-spar, 
and other important crystal types were elucidated, 
not only as regards general structure, but including 
the absolute magnitudes of their 7-values. 

Distinguished scientists, including P. Debye, 
P. Scherrer, P. Vegard, A. W. Hull, R. W.oWyekaas 
and L. W. Mackeehan, carried on the investigation 
of such atomic point lattices. 


Fic. 24.—Stereograms. Elementary cells in the fine-structure of crystals.} 
a. Fine-structure of copper, gold, silver, gold and aluminium. 4, Fine 
structure of rock-salt. c¢. Fine-structure of fluor-spar. d. Fine-structure 
of zinc-blende. ¢. Fine-structure of the diamond. /f. Fine-structure of 
iron pyrites. 


1 Some idea of the minuteness of crystalline lattice structure—of 
the diamond, for example—is obtained by a comparison such as the 
following : Imagine cubical boxes of 1 metre each side arranged one 
behind the other in a straight line from Berlin to Cairo. Now reduce 
this enormous length of nearly 3000 kilometres to 1 millimetre; a 
measure of the actual size of the elementary cells is obtained in the 
proportionally reduced boxes. There are about 2,837,000 of these 
to the millimetre, and a similar number of carbon atoms lie on 1 milli- 
metre of the edge of a diamond cube (Fig. 24e). Ina cubic millimetre 
of the gem there are 178 trillion carbon atoms. 

In considering the question further, it is noticed that in Fig. 24a the 


SIR WILLIAM HENRY BRAGG 


Professor of Physics in the University of London 


WILLIAM LAWRENCE BRAGG 


Professor of Physics in the University 
of Manchester 


Li) t 


CRYSTALLOGRAPHY, LEPTOLOGY 23 


If, however, it were immediately assumed that 
already a comprehensive idea of the microcosm 


copper atoms are to be thought of as being at the corners and at the 
centres of the sides of an elementary cube. Its length of side a is 
equal to 3°61 X Io ~° cm., i.e. 0:000,000,0361 cm. 

For silver this length amounts to 4:06, for gold 4:07, and for 
aluminium 4:07, all multiplied by 10o-% cm. The sodium atoms in 
rock-salt NaCl are arranged in exactly similar fashion (Fig. 245). 
The corresponding chlorine atoms occur at the mid-points of the 
Sigeseand atithe centre of the cube; @='5:8 X 10-°% cm. In 
Fig. 24c the calcium atoms of fluor-spar CaF, are placed in the same 
way as the copper and sodium atoms in Fig. 24a and 6 respectively. 
Now divide up the elementary cell into eight smaller cells by three 
planes, parallel to the sides and passing through the centre, and 
instal in each a fluorine atom. These atoms, owing to their positions 
at the centres of the cells, form a small cube inside the larger one ; 
a= 5°44 X I0—8, 

For zinc-blende ZnS (Fig. 24d) the metal is arranged as in the 
previous types, and like fluor-spar, the cubic structure is divided 
into eight compartments ; in this case only every other one contains 
a sulphur atom. The value of a for the zinc sulphide cube is 
5°4 Xx 10—5 cm. The structure of the diamond (Fig. 24e) is ob- 
tained from that of zinc-blende, if both the zinc and sulphur atoms 
of the latter are replaced by carbon atoms; a@ = 3°53 X to—& 
cm. Finally, iron pyrites FeS, (Fig. 24f) reproduces the earlier 
arrangements in the positions of the metallic atoms. Its sulphur 
particles set themselves on the diagonals of the eight smaller cells 
mentioned above, each one being about a quarter of the diagonal 
length from either the cube edge or the cube centre. Taken together 
the sulphur atoms form a rhombohedron, as Fig. 24f indicates. If 
the corners of this were moved along to the mid-points of the 
diagonals, the sulphur atoms would reproduce the arrangement of 
the fluorine atoms in fluor-spar CaF, (Fig. 24c) ; @ = 5°37 X 107° cm. 

It is necessary that the joining lines shown in the figure should 
be constantly kept in view. Moreover, it must be remembered that 
each atom represented by a point is itself a kind of planetary system 
with central body and satellites. 

Regarded in this way it is more than ever obvious that we have 
here a marvellous microcosmic system ; the particles hover in the 
crystal space like stars in the heavens, all mutually supporting one 
another in their regular arrangement, which we have learnt to 
measure accurately to a ten-millionth of a millimetre. 


24 CRYSTALS AND MATTER 


of the crystalline world had been obtained, our 
jubilation would be premature. Only a compara- 
tively small number of crystal stereochemical for- 
mule are available to-day, ten years after the first 
Laue research on the subject. This is to be attri- 
buted to the unsettled state of the times, and, in 
addition, to the indirect nature of the experiments.! 
Indirect methods must be used which, in complicated 
cases of atomic arrangement, are extremely difficult 
mathematically. 

This applies especially to the case of the com- 
pounds of organic chemistry.? 


1The atomic world is not disclosed by the microscope. As 
E. Abbé showed, the microscope is, in a certain sense, blind to objects 
smaller than about -o005 mm. (5 X 107-°cm.). The wave length of 
light visible to the eye, or of light such as can be used in photography, 
is coarse compared with the fineness of the leptons. The atoms are 
a thousand times smaller (only about 10~—* cm.), and can therefore 
no more act on light waves, which are large in comparison, than a leaf 
can influence the waves on which it floats. The wave lengths of X-rays, 
however (10~° to 10~* cm.), correspond well with such minuteness. 

A leptoscope using these fine waves has, however, not yet become 
possible, for no substance is known which would serve optically as 
a lens. We thus must rely on the original diffraction effect as in the 
case of the ultramicroscope with ordinary light. Probably at some 
time an image of the fine-structure will be obtained with X-rays 
by making use of the reflecting power of the structure planes in the 
crystal. M. Wolfke called attention to the possibility of separating 
in practice, as is done in theory, the formation of the image into 
the production of, first, a diffraction figure, and then, by further 
diffraction, the actual image. The production of the first diffraction 
figure would be assigned to the X-rays ; it would be photographed 
and then transformed by repeated diffraction, using ordinary light, 
into the image of the sub-microscopic object. With regard to several 
restricting conditions, those interested might read the ‘“‘ Physikalische 
Zeitschrift,’ vol. xxi, p. 495, 1920. The suggestion has not, so far, 
matured sufficiently to give practical results. 

* Recently W. H. Bragg, and also K. Becker and W. Jancke, 
have made some very welcome contributions to our knowledge of 


CRYSTALLOGRAPHY, LEPTOLOGY 25 


CRYSTALS AS STEREOCHEMICAL TYPES 


This being the case, it is obvious that, in the 
difficult work of establishing stereochemical formule, 
any external assistance is weleome. Crystal mor- 
phology makes its appearance in such a capacity. 
When carefully considered, this subject, in a certain 
sense, 1s merely macrostereochemistry. Of course, 
we must not conclude that a crystal is an enor- 
mous molecule similar to sub-microscopic molecules. 
That would be a false conception, being inconsistent 
with two actual properties of molecules, namely, 


Fic. 25.—Variety of forms of calc-spar. 


constant form and constant weight. In this respect 
we may compare the scheme for the benzene ring 
with the protean multiplicity of calc-spar (Fig. 25), 
and the exclusiveness of the molecule with the 
capacity of the crystal to grow and so to increase 
its magnitude and weight. A crystal form, however, 
is definitely characterised as a stereochemical symbol 
in the sense that it is a visible, and therefore easily 
examined, sample of the leptonic structure Its 


such substances. The dimensions of the elementary cells, and the 
number of molecules in each, are now known for indigo, anthracene, 
urea, succinic acid, hydroquinone, anthraquinone, naphthalene, and 
many other organic compounds, although the precise positions of 
the atoms are still uncertain, 


26 CRYSTALS AND MATTER 


principal surfaces represent series of planes through 
the leptocosm occupied by a network of atoms, and 
Fig. 26 shows clearly that in this way surfaces with 
rational axial sections arise; for the planes densely 


Fic. 26.—A plane in the space-lattice showing edges and surfaces. 


packed with matter form stable boundaries. The 
principal edges of a crystal indicate the directions of 
rigidly set lines of atoms, and its morphological 
symmetry is symbolic of the arrangement of its 
fine-structure particles. 


Fic. 27.—Rock-salt. Fine-structure Fic. 28.—Mirror symmetry of 
of the crystal faces and edges. Circles pyroxene. 
represent Na, dots Cl. 


The explanatory Fig. 27, which represents the 
known fine-structure of rock-salt, enables this to be 
clearly understood. In this way substances not yet 
investigated will be in great part determined from 
their external forms alone, and thus macrostereo- 


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CRYSTALLOGRAPHY, LEPTOLOGY 27 


chemistry will play the part which was emphasised 
as early as the year 1903 by G. v. Tschermak. 

In this sense Fig. 28, for example, gives us hints as 
to the structure of pyroxene, which 1s still unknown. 
Fig. 29, in the same way, provides definite indications 
as to the structure of that remarkable silicious 
material, quartz, for it exhibits in the crystal form, 
and therefore, we may assume, in its fine-structure 
(not yet experimentally investigated), not mirror, but 
only rotational symmetry, and serves as a macroscopic 
symbol of the famous conclusions of L. Pasteur con- 
cerning the “ asymmetrical’”’ molecule. In the crystal 


Mirror plane 


Fic. 29.—Right- and left-handed quartz. 


forms of quartz, actual right- and left-handed struc- 
tures are visibly and clearly characterised, and the 
compensated racemic variety also exists; this is 
shown in certain twin formations in which right- and 
left-handed quartz are combined in regular fashion. 

In addition, the remarkable property possessed by 
most crystals of splitting along definite planes affords 
an indication of the arrangement of the particles ; it 
may, of course, be assumed that the particles in the 
surface formed by a cleavage lie close to one another, 
each bound firmly to its neighbours. Perpendicular 


1 This is illustrated by imagining the two forms of Fig. 29 dis- 
placed parallel into one another. 


vik CRYSTALS AND MATTER 


to the cleavage plane, as in a direction of weaker 
cohesion, greater distances prevail, and in conse- 
quence splitting along such planes is possible. 


OUTLINES OF. GENERAL CRYSTALLOGRAPHIC 
MORPHOLOGY 


I. THE RELATIONS OF THE GROSS STRUCTURE 


In the way indicated above, the study of crystal 
morphology arises as an introduction to the subject 
of stereochemistry of the solid’ state. [t= isjiam 
this way, a part of chemistry. Every student of 
chemistry must, therefore, find himself immersed in 
a specialised subject at the outset. After a greatly 
changed cultivation of the crystallographic soil, much 
superfluous undergrowth of names and derivations 
having been rooted up and simple methods of 
development followed, the wanderer who picks his 
way carefully has no very special troubles to fear in 
this region ; indeed, it may be said that one is now 
able to go forth in this province, as in a well-tended 
garden, with artistic enjoyment. 

As an example of such a guiding plan in ae 
ornamentation of the inorganic world, so important 
in stereochemistry, the sans scheme will be 
briefly developed. 

The idea of deriving the raultisiticns of ei. 
graphic forms from five types all mutually connected, 
which may be called primitive forms, has already 
been brought forward by G. v. [schermak, and fol- 
lowed in his teaching. The fundamental rules of 
crystallographic symmetry and, therefore, of crystal- 
structural types, are embodied in these primitive 
forms, for they are fundamental in the varied mor- 


CRYSTALLOGRAPHY, LEPTOLOGY 29 


phology of crystals, being exemplified by the so-called 
centre of symmetry, axes of symmetry, and sym- 
metry planes. If the crystal structure possesses a 
centre of symmetry, to every boundary plane there 
belongs an equivalent parallel opposite surface, and 
thus lines through the centre of the crystal cut the 
external surface in two corresponding points. The 
symmetry axes indicate, in a certain sense, the 
rhythm in which similar structural particles grouped 
about a given direction repeat themselves in different 
positions ; thus if, say, the axis is senary, as in a six- 
sided prism, then this will appear to an observer, 
after a rotation of 60° about the vertical, just as 


Fic. 30.—The five primitive forms of aun pedion, pinacoid, 
sphenoid, doma, prisma. 

before. A symmetry plane divides a body into 

halves, which have the appearance of an object 

and its mirror image (Fig. 28, p. 26). 

The primitive form # (the pedion, Fig. 30) repre- 
sents a surface standing alone, i.e. a form devoid of 
symmetry ; #7 (the pinacoid), with a surface and 
a similar parallel surface, is the embodiment of 
the principle of centre symmetry ; s (the sphenoid), 
with one surface and another flap-lke surface, 
represents the fundamental idea of rotation in binary 
rhythm ; d (the doma) represents the principle of 
reflexion ; and finally, m (the prisma), the combina- 
tion d +s, d+ 1, or s + #1, which all give the 
same m. From these five types, to some extent 
representative of five structural rules in the inorganic 


30 CRYSTALS AND MATTER 


world, all the multiplicity of the remaining macro- 
stereochemistry may be derived by an application of 
the ideas of A. Schénflies as cases of rhythmical 
repetition of the primitive forms, according to the 
numbers 2, 3, 4, and 6, about a principal direction 
in the structure. Thus the ternary rhythm must 
appear in a whirl form and dominated by the octant 
(Fig. 31). It is fundamental that the rhythms may 
be developed in a simple (gyric) fashion, or by 


4 
N v 
Fic. 31.—Simple and isometric (octant) FIG. 32.—Gyric and gyroidal 


whirl structure. rhythms. 


rotation and reflexion together (gyroidal) (Fig. 32). 
Nevertheless, this development is very convenient, as 
the combined procedure leads to only four new forms. 

In the following table the thirty-two crystal classes 
are summarised as types of the crystallographic, 
and, therefore, stereochemical arrangements. These 
classes were known already in the time of J. Ch. F. 
Hessel (1831), and have all been discovered in 
crystalline materials, except in the case of 34. They 
are to some extent the architectural styles in 


a 


fr vat See oS Ly al 8. 
S3| 38) 3§ ao| 33 | 38 
KUE} aS | meg “EE | Ste] ges 
reed | RC) 4 Sag opel ob ics a | age 
ea ca} oa a A} ey 
Primitive forms: Triclinic and 
monoclinic system ™m co rms 
Binary rhythm of the primitive 
forms: Rhombic system . 2d 2m — — 
Ternary rhythm of the oS 
forms: Trigonal system 3a 3m 3p 35 
Tetrad rhythm of the rene 
forms: Tetragonal system 4d | 4m 4p 4s 


Senary rhythm of the primitive 


CRYSTALLOGRAPHY, LEPTOLOGY 31 


TABLE OF THE THIRTY-TWO CRYSTAL CLASSES. 


II. 


I. Gyric Development. Gyroidal 
Development, 


Structural Elements. 


forms : Hexagonal system 6d 6m — — 
Ternary rhythm of the primitive 

forms with octants: Isometric 

system ta zm — 


The rows are series of the same rhythm, the columns those of 
the same primitive forms. 2p and 2, being identical with m and s, 
are bracketed, and are only included in the table for the sake of 
completeness. _ 

The symbols of the classes are to be read, for example, as 
three p, three pi, and so or; 3 bar p, 3 bar s, etc.: written out in 
full they are, say, for the ternary series, ehgayte Pian trigyric 
pinacoidal, trigyric sphenoidal, trigyric domatic, trigyric prismatic ; 
sodium periodate, dolomite, quartz, tourmaline, calc-spar, are 
examples of these. The series is concluded with trigyroidal pedial 
(no example yet known) and trigyroidal sphenoidal (benitoite). For 
the isometric rhythm the development is characterised as isometric 
pedial, etc. 

Further details in F.:Rinne’s ‘ Einfuhrung in die Kristallo- 
graphisch-optischen sowie réntgenographischen Untersuchungen, ” 
4th edition, Leipsic ; published by Dr. Jaenecke. 


the crystal realm. A cursory examination of the 
table shows the simplicity of the relations running 
through it. 


32 CRYSTALS AND MATTER 


2. REPRODUCTION OF CRYSTALLOGRAPHIC FORMS 
IN PROJECTION 


It is especially convenient in considering the 
matters treated above to turn from pictures of the 
crystallographic configurations in perspective and 
to symbolise the formal relations in projection. In 
Fig. 33 this is done for the primitive forms. To 
render the figure clear, it is noticed that an X in the 
diagram represents a surface in the upper portion of | 
a crystal which has been cut in half by a horizontal 


FIG. 33..—Projection diagrams of the five primitive forms of crystallography. 


Fic. 34.—Projection diagrams of the senary rhythm of the primitive forms. 


plane ; a circle represents an under surface. Such a 
“figurative point ’’ in the centre of the projection 
denotes a surface parallel to the paper, whilst one on 
the circumference represents a plane perpendicular 
to the paper, meeting the circle in the point marked. 
Finally, the points between the centre and the cir- 
cumference are symbols of surfaces placed obliquely. 

It is now easy to represent the effect of a rhythmic 
repetition of the five primitive forms in projection. 
In Fig. 34 this is done for a senary rhythm, using 
oblique surfaces as the general case. The result gives 


CRYSTALLOGRAPHY, LEPTOLOGY 33 


the five simple whirl elements of the hexagonal 
system. A development in accordance with the 
principle of rotation-with-reflexion gives nothing new 
in the hexagonal series. In a similar manner we 
obtain in the other systems general examples of the 
external crystallographic forms, and at the same 
time indications as to the arrangement of the particles 
in the fine-structure. 


3. FINE-STRUCTURE RELATIONS OF GENERAL 
CRYSTALLOGRAPHIC MORPHOLOGY 


For the purposes of leptology, this grouping in 
crystal classes is further subdivided into a large 


er Refi exton- 
Simple Simple with- 
yoluion atin peheniGn translation 
Trigonal Trigonal Mirror plane. Plane of reflexion- 
rotational serew with-translation 
axis axis 


Fic. 35, a-d.—Simple rotation and translation-with-rotation (screw form). 
Simple reflexion and translation-with-reflexion. 
number of space groupings, thus completing the 
classification. As new variations of the symmetry 
elements, we have to consider the following: transla- 
tion-with-rotation and translation-with-reflexion, that 
is to say, the resultant motions arising from rotation 
or reflexion, together with translation. The result in 
the former case is a screw form. Fig. 35) serves as 
an example of such a screw axis, and Fig. 35d shows 
a translation with reflexion. 
For the rest, it is recognised as a result of the 
3 


34 CRYSTALS AND MATTER 


space-lattice principle, that for every crystal struc- 
ture, considered from a stereochemical standpoint, all 


Fic. 36.—Asymmetric fine- Fic. 37.—Fine-structure with 
structure. symmetry centre. 


Fic, 38.—Fine-structure with Fic. 39.—Fine-structure with arrange- 
binary arrangement. ment on a binary screw axis. 


Fic. 40.—Fine-structure with Fic, 41.—Fine-structure with trans- 
reflexion. lation-with-reflexion. 


Fics. 36-41.—Simple stereograms of crystallographic fine-structure. 


the symmetry elements, and therefore all the fine- 
structural particles, occur in the crystal in periodic 


CRYSTALLOGRAPHY, LEPTOLOGY 35 


sequence, as Figs. 36-41 show for the simplest cases 
of crystal structure. In short, allowing for all the 
crystallographic possibilities, there are 230 types of 
crystallographic fine-structure, and consequently a 
similar number in the stereochemistry of the solid 
state. Every crystallographic substance is consti- 
tuted according to one of these schemes of A. Schén- 
flies and E. Fedorow, and its specific character in the 
material world is expressed in the absolute measure 
of the periodicity in the point system, and in the 
magnitude of its angles. 


EXTENSION OF THE MANIFOLD OF CRYSTALLO- 
GRAPHIC TYPES BY TWIN FORMATION 


Reviewing the fine-structural relations, the thirty- 
two classes with their subdivisions into 230 space 
groupings present themselves as the embodiment of 
the principle of association. Each crystal unit is the 
model of a structural style containing certain elements 
of this aggregate. Taken together they constitute 
a complete system of forms based on Hatiy’s funda- 
mental law (p. 8), a system to which mathematical 
thought has nothing to add. 

It is thus a surprising thing to observe that nature 
in numberless cases pushes this principle of associa- 
tion still further, for she takes as unit the whole 
crystal and combines it in regular fashion with its 
like to give so-called twinnings. It was G. v. Tscher- 
mak again who pointed out the parallelism between 
form development from elements, such as the five 
primitive forms and this higher association into 
twins. If a certain crystal lacks a centre of sym- 
metry, as, for example, the “‘ hemimorphic succinic 
iodimide ’”’ shown in Fig. 42, this is occasionally 


36 CRYSTALS AND MATTER 


remedied in nature by a regular growing together of 
two individuals. The complex (Fig. 43) is centre 
symmetrical. In the same way, twin formation with 


FIG. 42.— FIG. 43.— FIG. 44.— 
Hemimorphic crystal of Centre symmetrical twin Right-handed 
succinic iodimide. of succinic iodimide. quartz. 


respect to a symmetry axis is accompanied by an 
increase in the symmetry. If, for example, a right- 
handed quartz (Fig. 44) exhibits the symmetry 
elements 3s, then its association, often occurring as 


Fic. 45.—Left-handed quartz. Fic. 46.—Right- and left-handed 


quartz, twinned. 


an ingrowth with a left-handed quartz in a twinning, 
‘as in Fig. 46, is of the type 3m. Thus there is 
an increase in the symmetry from the sphenoidal s 
form to the prismatic m form (p. 31). On the other 


CRYSTALLOGRAPHY, LEPTOLOGY 37 


hand, a regular twin formation of two right- or left- 
handed quartz crystals (Fig. 47) has the symmetry 
6s. Such regular non-parallel combinations can be 
constructed by rotating one crystal with respect to 
the other about a fixed axis (in the example of 
Fig. 47, 60° about the whirl axis). A combination 
along a twinning plane z, after application of a rotation 
of 180° about the normal to z, acts similarly as illus- 
trated by gypsum (Figs. 48 and 49). 


Fic. 47.— Fic. 48.—Gypsum. Fic. 49.— 
Two left-handed quartz, Gypsum twinning. 
twinned. 


The symmetry is thus augmented by a mirror 
plane, and the type 2d replaces that of m. 

Such extraordinarily widespread phenomena in 
nature are certainly of considerable general signific- 
ance. This consists primarily in the recognition that 
a visibly high symmetry may in reality represent an 
ageregation of parts of lower symmetry, even if in 
the limit the complexity is no longer recognisable 
by the eye owing to fineness of the parts. Actually 
there are many gradations of such mimesy from 
macroscopically definite twinnings (such as pseudo- 
hexagonal chrysoberyl) (Fig. 50) to the finest struc- 
tures arising from “‘ polysynthetic’’ repetition of 


38 CRYSTALS AND MATTER 


very thin lamellz, shown, for example, by microcline, 
the fineness of which verges on the sub-microscopic. 

We may recognise, therefore, in twin formation 
an attempt towards higher symmetry than that 
possessed by the individual crystal; in addition, 
however, the relations also indicate that the twin 
grouping, in contrast to the ideal parallelism in the 
structure of a simple crystal, presents a less compact 
fitting-in of the particles with each other. The space-. 
lattices of the parts of the twin do not pass con- 
tinuously one into the other. In this respect it is 
of further interest that, according to the ideas .of 
Ch. Friedel, fine-structural parallelopipeds, at least 
of higher orders, may be constructed which satisfy, 
or nearly so, the condition of parallelism. The 
diagram of Fig. 52 makes this clear. 


Fic, 50.—Chrysoberyl sextet as an Fic. 51.—Fine twinning of microcline 
example of mimesy (micrograph) 


ie 4 A | 


i Wy 


fi cite il Hf 
Aas Ff ease 


Fic. 52.—Fine-structure of a twinning 


.s Tile LIBRARY. 
OF THE 
URIVERSITY oF jLLINOES 


IV. FINE-STRUCTURAL UNITY OF MATTER 


FINE-STRUCTURE OF AMORPHOUS BODIES COMPARED 
With FHAT OF CRYSTALS 


were to leave the fine-structural relations of 

crystals, which have chiefly interested us up to 
now, without attempting to connect them up with 
general leptology which concerns the physicist and 
chemist, in their study of the numerous examples of 
amorphous bodies. Actually, crystals and amor- 
phous bodies are very closely related regarding their 
fine-structure. The name of the latter class hardly 
accords with the facts of leptology. The ordinary 
chemical formule and ideas concerning the mor- 
phology of atoms of gases and liquids, indicate 
clearly a definite anisotropic structure. In this way, 
chemical formule, such as, for example, 


/ CoH, N(CHs)2 ; 
NaCl; (NH,) Cl; C,H, C 


I; would be contrary to scientific principle if we 


a 
PG, N(CH,)2 


NH, | NH, 


Cl Ni Cra bis roa CO rs, 


NH; | NH; 
NH, 
and the diagrams of Fig. 53 represent fine-structural 
schemes: To F. M. Jaeger must be accorded the 
special merit of having first recognised and enlarged 
39 


40 CRYSTALS AND MATTER 


upon the symmetry relations of the individuals of 
amorphous bodies in his very fine work, “ Lectures 
on the Principle of Symmetry” (1917). Figures such 
as 53, 54, and many others of atoms, ions, and mole- 
cules show, without further remark, a regularity in the 
form of the individual leptons, a regularity a 


O : 


Electron H: Ion 
ca. 10-13 cm. ca. 10-16 em. OR ee Atom ca. 0°55 X 10-8 em. 


nae 


A, Molecule Crystal 


Fic. 53.—The leptonic series. Models of an electron, atom, ion, molecule, and 
a crystal. 


d Inosite t Inosite 


ef 
eH 
°OH Symmetry plane 


o86 


Fic. 54.—Examples of symmetrical molecules. 


referred to by chemist and physicist. Considerations 
of symmetry, whether it be total absence of sym- 
metry, centre symmetry, rhythmic architecture in the 
sense of gyric or gyroidal repetition, or, finally, mirror 
symmetry, are of importance here as in crystals. 

In fact, we may assert with P. Debye and 
P. Scherrer that, allowing for the difference of scale, 


UNITY OF MATTER 41 


as regards general structure there is no essential 
difference “‘ between a crystal and a chemical mole- 
cule, for both have the characteristic property of 
containing atoms regularly arranged.” 

With respect to this intimate relationship, it 
may be added that between the leptonic forms of a 
liquid or gas from which crystals are separating and 
the crystals themselves there is a definite morpho- 
logical connection. To each crystal type will belong 
certain characteristic preliminary forms occurring in 
crystallisation. 

This fundamental and general relationship of the 
fine-structures of atoms, molecules, and crystals may 
also be applied to the arrangement of the negative 
electrons forming the external shell of the atom, 
which swarm of peripheral corpuscles gives to the 
atomic complex its “shape.’’ We speak of the 
symmetry of atoms as of crystals, meaning by that 
the symmetry of the external electron arrangement. 
The tetrahedral form of a carbon atom, for example, 
is indicated by four electrons at the corner of a 
tetrahedron. Following A. Johnsen, we presuppose 
a minimum symmetry of the atom in crystallo- 
graphy, and this is determined by crystallographic 
symmetry relations. The minimum symmetry for 
a given kind of atom changes with the symmetry 
conditions prevailing at the place where the particle 
occurs in the fine-structure. The carbon atom of 
the diamond has to fulfil other symmetry relations 
than that of graphite. In consequence, the arrange- 
ment of the electrons in these two cases must be 
different. That this is so is shown distinctly by their 
different optical properties. Thus we may compare 
the transparency of the diamond, on the one hand, 


42 CRYSTALS AND MATTER 


with the blackness of graphite, on the other. But 
even in the same crystal the electron arrangements 
of similar atoms may differ. In the diamond all the 
carbon atoms are identical ; in graphite two varieties 
must be postulated to satisfy symmetry considera- 
tions. Recognising this, the mutual influence of 
different atoms and ions in the crystal is to be borne 
in mind, for their forms must depend on their neigh- 
bours for the time being. The action of a fluorine 
ion on the fine-structure of a neighbouring sodium ion 
is different from that of chlorine, bromine, or iodine 
ions, and these again are affected differently by the 
electron orbits of sodium potassium, rubidium, and 
caesium. Moreover, temperature itself must be recog- 
nised as a factor tending to determine the structure. 
The electron arrangement in free atoms or mole- 
cules will not be essentially different from that in 
the crystal. Indeed, recently the physicists M. Born, 
H. Landé, and others, following crystallography, 
speak of cubic and other polyhedral atoms. The 
diagrams in Figs. 55-57 are drawn in accordance 
with this general idea. In particular, the known 
periodicity of the series of the elements with respect 
to the number eight, suggests as a probable distri- 
bution a surface-centred octahedral arrangement for 
elements with eight external electrons. By increasing 
the volume, that is, by adding further electron shells 
for the elements of higher atomic number, other 
stable arrangements, partly of a crystallographic 
nature, but approximating to an isotropic distri- 
bution, will arise; such configurations are treated 
in detail in a recent treatise by H. Tertsch.t Fin- 
ally, the electron swarm becomes less stable as 


1Compare p. 84 and p. 181. 


UNITY OF MATTER 43 


the charge on the nucleus is increased, so that the 
series of possible elements on our earth ends with 
uranium, a spontaneously 
disintegrating atom with 
ninety-two external elec- 
trons. 7 


© Nucleus 
electron 


ata 


e Nucleus with two electrons 
© Hlectron 


Fic. 57. 


Fics. 55-57.—Schemes for electron groupings in atoms, molecules, and 
crystals. Examples: carbon atom, methane molecule, diamond. 


PHYSICAL INVESTIGATIONS ON THE GENERAL FORMS 
OF ATOMS AND MOLECULES 

Fitting in with our ideas on the analogy 
between crystals and individual leptons, a similar 
correspondence in the behaviour of crystalline and 
amorphous substances of the same chemical com- 
position is shown in certain physical processes. We 
find that the curves obtained by Cl. Schaefer and 
M. Schubert for the reflexion of short ultra-red 


44. CRYSTALS AND MATTER 


waves by quartz and opal completely correspond, a 
clear indication of the fine-structural similarity of the 
SiO, particles, in spite of the difference in the gross 
forms, quartz showing the characteristic space- 
lattice structure lacking in the opal. In their action 
on X-rays, too, amorphous and crystalline materials 
behave in much the same 
general fashion. An ex- 
tremely fine crystal pow- 


| der approximates for this 
4 --f---4 purpose to the amorphous 
Oe eee material with which the 


2,4 6) 8 ORR 8 ee eey complete crystal 4S een 
ultra-red for quarts and opal. After linked up through inter- 
Cl. Schaefer and M. Schubert. mediate forms. Accord- 
ing to Debye and Scherrer, the subdivision in soot 
actually extends to complexes of only thirty atoms of 
carbon; but these still give the normal X-ray effect 
of crystals. 

With amorphous substances circular shadings 
round the point of impact of the primary ray are 
obtained, although, on account of the small number 
of components in the kinetic unit, the considerable 
internal heat motions, and the irregular state of 
aggregation! of the particles, only one or two weak 
and diffuse rings appear on the plate. It seems to 
me, however, that all this points to a definite regular 
form of the particle. Fig. 59 illustrates the produc- 
tion of such an effect, exaggerated for the sake of 
clearness, with the imaginary case of a molecule 
with cubic arrangements of its atoms.’ 


i 
SCC Neel 


1 Also occasionally, as in glass, owing to the interaction of the 
different types of molecules. 

* For readers interested in crystallography, diagrammatic deriva- 
tions of the lines in the Debye-Scherrer diagram are shown in Fig. 59, 


UNITY OF MATTER 45 


NotE.—The conception of colloidal matter is not 
synonymous with that of amorphous substances. 
Its characteristic 1s a medley of particles of magni- 
tude 10-* to 107-7 cm., which thus lie between 
leptonic dimensions (about 10~® cm.) and micro- 


Ses Phot. plate section) 
Phot. film (cylinder) 


iP 


| 
: 
oe Se ae 


Incident Beam Incident Beam Incident Beam 


geass 
meas, 


Fic. 59.—Debye-Scherrer diagram. Example of a cubic molecule shown 
schematically. 


scopic visbility (about 10-5 cm.); they may be 
either crystalline or amorphous. The colloidal solu- 
tions of gold and silver, for example, previously re- 
garded as amorphous, show, according to P. Scherrer, 
obtained from the principal structure planes in the imaginary cube 
molecule. These are obtained by applying the principle of reflexion, 


but it must be remembered that the fundamental phenomenon is 
one of diffraction. 


- 


46 CRYSTALS AND MATTER 


the typical phenomena of crystalline materials (Fig. 
60, upper diagram). Silica gel also gives a crystal 
effect, but at the same time acts an amorphous sub- 


Fic. 60.—Debye-Scherrer diagrams of colloidal silver and silica gel. After 
P oeherrer. 

stance (Fig. 60, lower diagram). In this preparation 

we have to deal with an amorphous gel containing 

small SiO, crystals disseminated in it. 


ATOMIC DOMAINS 


Stereograms of crystals, such as that of Fig. 61, 
which shows the elementary cell of metallic sodium 
in representing the atoms as mere points, give no 
indications as to their solidarity.t It is possible, 
however, to endow such symbols of the crystal 
structure with a dynamical basis, and, in fact, this 
is done by describing about each atom a domain, to 
represent a portion of space which it claims for 
itself and keeps free from other atoms. Under the 
action of attractive and repulsive forces the atoms 
in the structure mutually maintain each other in an 
equilibrium arrangement, which is characterised in 
the close packing of these atom domains, which are 
to the first approximation spherical. The radius of 
the domain is given simply as half the least distance 
between the atoms in a stereogram similar, say, to 
Fig. 61. Thus Fig. 62 shows the domain picture for 
the sodium of the previous stereogram. 

In this case the radius of the domain of each atom 


1 In the scheme for diamond (p. 43) this is indicated. 


UNITY OF MATTER ~— 47 


amounts to 1°86 x 10~*® cm. Passing from one 
material to another as from sodium to sodium chloride 
(Fig. 63), thus comparing simple 
substances with compounds, we 
arrive finally at a table of magni- 
tudes for the atomic domains, a 
table obtained by W. L. Bragg, who ~ 

first carried out systematic experi- Ti 

ments on these lines. Fic, 61.—Stereogram 

P. Niggli has given a similar  °% scum cyst. 

scheme. Further applications in crystallography are 
due to G. Aminoff, and the author has also expressed 


Fic. 62.—Atom domains in stereo- Fic. 63.—Atom domains in stereogram 
gram of sodium crystal. of rock-salt. 

his views on these very promising relations, both in 

his lectures and in occasional publications. Some 

new determinations may here be put before the 

reader for his guidance (compare also p. 107). 
Diameters of crystallographic atom domains in 

10 ® cm. units :— 


E35 ane ees-OOulbiger e. ”.|-2-69 \ = Pe Me is ty ine, 1+26 
eas. epenine are vg. “Sct Cla Ae ae ot Rea a . 2°00 
| he eae ort 9 45 3°90.) Bre Jae Ae ANC. pea 
Bpre" may Bart a e436}. ee eg me owe so 
Cac. » 5°04 


That the atomic domain depends on the nature of 
the surrounding atoms and the electric charge, has 


48 CRYSTALS AND MATTER 


already been emphasised by K. Fajans, H. Grimm, 
and W.L. Bragg. The alkali metals, for example, 
show somewhat larger values than their salt ions 
tabulated above. We have, in fact, Li, 3:02; Na, 
B7 ER NS ASO WARD M4 NOS Ue 1on 

The transference of these numbers obtained in the 
case of the crystal to the individual leptons, is a 


Oo) 02) 
Co ee 


Brides + dno coco 
RTT Co 


"290 ‘aekie. eee 
Fic. 64.—Molecular domains Fic. 65.—Molecular domains of 
OF 0 guNor CO. CO)., iH PEIN 35 70 


procedure in perfect accord with the views adopted 
in this book. In accordance with the considerations 
on p. 39 et seq., the chance of error is not so great as 
might at first appear. An investigation of this point 
is contained in the Figs. 64 and 65, and the values 
given agree very well with those arrived at from de- 
terminations of the mean free paths of the molecules. 


Molecular Diameter. 
Crystallographic Data. 5 : : 
y Hed Landolt and Bornstein’s Nernst. Theoretical 

Tables. Chemistry. 
O, ued §2) eek De ems ae 00 107~*% cm. | 2:00, 4.) a0ssee 
N, os 2600, rOp Bem. S80 400) NO 3 cms ao 7 eee 
GO, 2s BOO rs TO Orca 2-9 a. 1G Sem eo) ee 
Cl, eo 4240-S°> 2107 cm.)| 48) | SIO om ara A 


The mathematical treatment of the question leads 
to the most difficult branch of mathematics, the 


UNITY OF MATTER 49 


problem of several bodies. Fora long time, however, 
approximations must suffice. Nevertheless, thanks 
to the efforts of M. Born, A. Landé, K. Fajans, 
F. Madelung, H. Thirring, and others, methods of 
approach to the end in view have been laid down, e.g. 
with respect to the attractive and repulsive action 
of the ions. The attraction here is put inversely as 
the square of the distance apart of the ion centres, 
whilst the potential of the repulsive forces (corre- 
sponding for the alkali halides to the compressibility 
of the crystal) involves a higher (5-9) power of the 
distance. This will naturally depend on the size and 
nature of the particles, and in view of the anisotropy 
of the sphere of action, may vary with direction. 

Although the working out of exact mathematical 
theories of the fine-structure of matter must be left 
to the future, a glance at the spacial crystal schemes 
at once suggests the diagnosis and interpretation of 
many peculiar properties of substances. 


DIFFERENCE BETWEEN THE STRUCTURE OF 
INDIVIDUAL LEPTONS AND CRYSTALS 


The chief difference in the construction of crystal- 
line bodies and the individuals of amorphous sub- 
stances lies in the restriction of the rhythm. In the 
fine-structure of crystals, owing to the fact that 
each structural unit is joined to its neighbours, 
this rhythm exhibits a three-dimensional periodicity. 
Such space-lattice structure is clearly only possible 
when the repetition is according to the numbers 
2, 3, 4, and 6, or with no repetition at all. Pentad, 
septad, and compound axes of higher period are here 
theoretically excluded, nor are they found in practice. 

4 


50 CRYSTALS AND MATTER 


A boundary surface not containing rifts or gaps is 
known to be impossible with such polyhedra (Fig. 66) + 
We thus have here the essential principle in the 
restriction of the crystalline forms to the thirty-two 
classes of page 31, and thus to 230 space groupings. 


Crystallographic Rhythms 


FECOOHH 3 


Digyral Trigyral Jetragyral Aterogyraf 


Non-crystallographic rhythms possible in molecules 


Fic. 66.—Diagrams showing crystallographic and two non-crystallographic 
rhythms (pentad and septad). 


2 ted 2 cag 
Fic. 67.—Irrational axial section for eight-fold rhythm. 


Further, this fine-structural limitation is in agreement 
with Haitiy’s crystallographic law of simple rational 
axial sections. A crystal rhythm corresponding to 
the number 8 (Fig. 67), cannot arise, for such a 


1F, A. Wilfing has already pointed out the importance of this 
fact in crystallography. 


a 


UNITY OF MATTER 51 


regular octagon would give an axial section of 
2°4142 ... Similar relations hold good when repeti- 
tion occurs according to any number other than 
2, 3, 4, or 6, from which it also follows that of the 
“yegular’”’ polyhedra of mathematics, the cube, 
tetrahedron, octahedron, and icosahedron, only the 


BOY 


Fig. 68. Fic. 70. Fic. 7I. 

Fics. 68, 70, 71.—The mathematical regular polyhedra of crystallography. 
first three are represented in the inorganic realm of 
nature. The Figs. 68, 70-72 should show the reader 
that the first three fall within the above restriction 
in the rhythmic arrangement of their surfaces, whilst 
the icosahedron, the non-crystallographic five-fold 
repetition of which is clearly shown in Fig. 72, does 


Fic. 72.—The non-crystallographic regular icosahedron. 


not. Such inadmissible rhythms are absent from the 
macroscopic crystal form, and also from the fine- 
structure. With regard to the latter, the Laue dia- 
grams are unimpeachable witnesses (Fig. 73) ; in the 
structure of free atoms and molecules, both crys- 
tallographic and non-crystallographic rhythms may 
equally well occur. 


NoTE.—Fic. 69, which appeared erroneously in the original, has been 
deleted. 


52 CRYSTALS AND MATTER 


e eee . om. a nie Pe . . 
ee a H 
2 ae & or o Sea a : 
. wey Ay Os Se e e e ie A ee y i Py 
e Ye Pe ee a2? © ® ®@ e 5 . aOR oi" 6 age e 2 sree 
bo dK) *e, ® °@e- 5 ‘°°? “ ris 4 e Ae O°8e" 0. : e 
a e 6 ‘ ie? eco Ps ae) or or Py eg es a ee ° 
oe f is . . @ «= %6 
eee oe ae st sees See 
® e. @ e of e ~ “2 Cn ) e-.° e 
ee Orne Yee ° ° ve. ce Na 
. * ee © e ee ei iY ete G e Par e nies © i. ae 
° Ay ae sad oS ® e° e seek at eer ae’: a 
ee 4h ote g co's & e ee. . Xe * ae $ ee, = x 
® @ Cae, ett te a Ai 
e ® Seg? e Vee, ene Noy ee : 
e eel we By ° @ e a9 Sb ars ® =e 2 wee . 
76°. hg 29g 28 - ° pite* os) . eee Ce0-6 <7 Yea oe 
@ a Si\e°8 Se e res 5 acti sie eee e wee 
e e& eo? ° e ® oF. e - oO se: AO ae 
e oS @ee SP a! e e e See ae 5 at 
° a 3 ; 
ee, 2 ae labes ; 
° e e o%8 c & « : - : a 
e e F 
= b 
: y ; . % . 
e hd a ; 
i xs e. “ 
wae, ? ; e e. . 
x ; ad 5 ° ° e @ ° 
° e e e ° te eo ®@ «6 . 
ee .- @& e e e . ; : 
hese Ms ry} Pig e° . e e e corie Ab e @ Rn 
s . : . e OF , m 
rey ADRS e ~* ® oe? - 3 we ky) ; ay, 
° ate . or ae? de ° n A ? 
eee ein Want | ana) oe e € ; 
e we ee Ayo e e e ° e e 
3 a A . e ; e . e .° e, z 
? S e. ret s . ‘ : : 
° -@ ° ts ‘ e” ~° ®@ 
' 3 Were : ae 6 +3 3s e, e ° 
om 6 ~® e e oe ‘ @ e Q a e : 
ee ° ee ® e e@ 6¢ e ae 5 as 4 Pd ‘ / 
¥ e Chk ert PuAP ar peer) v3 - e e . 
e aa) ore e were Z ° . e ) e 3 
e mh pee e e, . ( ses , >| 
se : - . ° 
i! Pas ° e ang 7 ; 3 2 
e e e e 
e e 6 e . . 
° e 


rj oo ’ 

e*e. ,e° Sey ote ey °@, Jee 
eee 

O10 see Pov ." 9. 8e® 2 Shia y 

: 


Fic, 73.—Laue diagrams with repetition according to the numbers 1, 2, 3, 4, 


and 6. After photographs by F. Rinne. a. Cyanite. b. Sanidine. 
c. Calc-spar. d. Rock-salt. e. Beryl. 


UNITY OF MATTER 53 
DIAGRAMS 


Binary type. Rhombic system. 
Ternary type. Trigonal system. 
Tetrad type. Tetragonal system. 


Senary type. Hexagonal system. 


Fic. 74.—Whirl forms of the crystal types deduced from the primitive forms. 


V. THE GENERAL CHARACTERISTICS OF 
THE FINE-STRUCTURE OF MATTER 


FINE-STRUCTURAL CHARACTER OF CRYSTALS 


F, as we have emphasised, the difference in the 

architectural rhythms of the structures of indi- 

vidual leptons and crystals is merely a matter of 
detail, then the question arises as to what constitutes 
the prevailing characteristic in the fine-structure of 
all substances. 

Taking crystals as types, we find that they ex- 
hibit very definitely two peculiar qualities-—firstly, 
change of properties with direction (anisotropy) ; 
and, secondly, stability (isostasy). 


A. CHANGE OF PROPERTIES WITH DIRECTION IN 
CRYSTALS 

This anisotropy makes its appearance in the mor- 
phology of the crystal, in the regular arrangement 
of the external faces. Rock-salt, for example, may 
develop in a certain direction a surface, say, that 
of the cube; and this development repeats itself 
in a definite series of isolated directions, to which 
the edges and angles of the structure conform. 

The diamond is provided with an octahedral 
form in a similar way. As to the fine-structure, 
Figs. 75-78 represent with remarkable clearness the 
various modes of formation of certain principal 

54 


FINE-STRUCTURE OF MATTER 55 


surfaces of the minerals, showing the anisotropy of 
the structure. 
| The change of physical properties with direction 
is very obvious in crystals showing cleavage. Thus 
for rock-salt there are three directions, perpendicular 
to the cube surfaces, in which the cohesion of the 
crystal is a minimum. The resistance to splitting 
in these directions is only one-third of that in the 
direction of the cube diagonal. The cleavage planes 
are, therefore, regularly oriented surfaces of maxi- 
mum brittleness. In like manner, many crystals 
show particular planes of maximum plasticity. These 


Fic. 75. Fic. 76. Fic. 77. Fic. 78. 


Fics. 75-78.—Structure of the diamond parallel to surfaces of the cube, rhombic 
dodecahedron, pyramidal cube, and ‘octahedron. 
are surfaces in which internal displacements may 
easily occur; with ice they arise as planes parallel 
to the surface of the ice floe. Hardness is also a 
directional property in crystalline materials. The re- 
sistance to disrupture, which shows itself as hardness, 
is often different in different directions of the crystal ; 
thus it is a familiar fact to diamond workers that the 
cube surfaces of the gem are more difficult to prepare 
by polishing than the octahedron surfaces. Garnet, 
too, is harder on the cube surfaces than on the 
octahedron and rhombic dodecahedron, according 
to the researches of P. J. Holmquist on polishing. 
Even on the same surface of a crystal the hardness, 


56 CRYSTALS AND MATTER 


as measured by scratching, varies with the orientation 
of the scratch made by the test needle. Cyanite is 
a classical example of this. 

Further, demonstrations of this change of pro- 
perties with direction are given, often very strikingly, 
by optical tests. On passing ordinary daylight 
through the mineral cordierite such extremes as 
these arise in the absorption of light ; a preparation 
in a certain direction appears dark blue; that in 
another direction, yellow; and in a third, grey. | 
Moreover, the customary idea of constant wave 


Anisotropy of Crystals 
(Regular change of properties with direction] 


oS 
: 
ee 2 
ae A | right blue | P= 
eZ a8 
yx 
Anisotropy Anisotropy of Anisotropy of Anisotropy of Anisotropy of 
of form Cohesion, for Lightabsorption Heat Conduction Ghemical Reaction 


Example: Rock-salt. example: Cleavage. Example: Cordierite. Example: Gypsum. Example. Tourmaline. 
Example: Mica. 


Fic. 79.—Demonstrations of the anisotropy of crystals. 


length as a measure of the velocity of propagation 
of the light fails for all non-isometric crystals. For 
such substances the value of X for a given colour 
changes with the direction of the light, and wave 
length curves can be drawn showing the variation 
diagrammatically. 

Variation of thermal properties with direction 
may also, in many crystals, be very clearly demon- 
strated. Fig. 79d indicates the heat conduction in 
a cleaved plate of gypsum. A thin layer of wax has 
been deposited on the gypsum plate, and the prepara- 
tion heated from a central point with a hot wire. 


FINE-STRUCTURE OF MATTER 57 


The wax is melted to various extents in different 
directions corresponding to the heat conduction in 
gypsum ; the perimeter of the figure formed by the 
melted wax is indicative of the heat conduction in 
the crystal beneath. 

Even in chemical actions a definite change of 
properties of materials with direction is unmistak- 
able. Thus, for calc-spar, varying intensity in the 
reaction . CaCO, + 2HCl = CaCl, + H,O + CO, is 
clearly shown by the different amounts of CO, 
liberated under the same conditions from the different 
. faces of the mineral. Hence resistance of calc-spar 
to attack by hydrochloric acid varies with direction. 
The differences which arise are surprisingly large. 
According to O. Miigge, quartz is attacked by fluoric 
acid 150 times more easily in the direction of 
the whirl axis than in the direction perpendicular 
thereto. 

It is of considerable significance that in crystal- 
line materials “ simple vectorial’ directional differ- 
ences can occur. The tetrahedron is a morphological 
example of this; it lacks a centre of symmetry. 

A chemical example is depicted in Fig. 79e. The 
diagram represents a tourmaline sphere which has 
been transformed by caustic potash to a bee-hive 
shaped body, a clear indication that the chemical 
reaction between the silicate and its corrodent occurs 
much more rapidly in the direction from below to 
above than from above to below. 

It is, therefore, a characteristic of crystalline 
materials evidenced by numerous experiments of 
morphological, physical, and chemical nature that 
they exhibit different properties in different direc- 
tions. 


58 CRYSTALS AND MATTER 


Corresponding to this the sphere of action of 
crystals is also anisotropic, as growth phenomena in 
particular show. A crystalline sphere grows to a 
body with edges, corners, and plane faces. 


By SITARILIIM OF CRYSTALS 


Associated with the characteristic of anisotropy 
in the crystal is the second general property, internal 
equilibrium or isostasy ; the whole constitution of — 
the crystal as a stable form exemplifies this. A. Nold 


Benzene molecule 


Fic. 80.—Morphological anisotropy of molecules. 


has especially taken this into account in his work 
on the crystal structure of the diamond. However, 
in the fine-structure of crystals the question is one 
of dynamical equilibrium, that is to say, kinetic 
stability or isodynamostasy.* 


1 As a parallel to this, there is the isostasy of geology, from which 
science the name isostasy is here borrowed. The earth, however, is 
not in isostatic, but in isodynamic equilibrium. Its rotation in- 
volves equatorial bulging, and as a result of this there arise fissures 
tending to prevent such an adjustment ; these, as the boundaries of 
continents, and rifts within them, run from north-east to south-west, 
south-east to north-west, and meridionally. 


FINE-STRUCTURE OF MATTER 59 


FINE-STRUCTURAL CHARACTER OF GASES AND 
LIQUIDS 
The general characteristics in the construction 
of the individual leptons (electrons, ions, atoms, 
and molecules) cannot be different. The graphical 
schemes and formule given for the structure of atoms 
and molecules indicate morphological anisotropy ; 


as 


Benzene drop 


Fic. 81.—Pseudoisotropy of an aggregate of molecules (benzene drop). 


but their morphological, as well as their chemical and 
physical anisotropy, will be entirely obscured by the 
irregular arrangement of the particles and will thus 
be transformed into an isotropy by averaging. Mor- 
phologically, this is illustrated by the spherical form 
of free gases (e.g. the earth’s atmosphere) and in the 
drop shape of liquids. The general stereophysical 
conception of the electrons, atoms, and molecules as 
kinetic units at once indicates their isodynamostatic 
character. 


60 CRYSTALS AND MATTER 


GENERAL CHARACTER OF THE FINE-STRUCTURE OF 
MATTER 

According to the above, the fine-structural ar- 
rangements of every substance represent anisotropic 
stability forms. 

Their structure will be conditioned by attracting 
and repelling forces. The particular arrangement of 
the particles of an aggregate is always the result of 
a complex action of all its particles on one another; 
it is not characterised by some kind of linear or 
curvilinear force threads. 

In addition, the aggregation units, whether atoms, ~ 
ions, molecules, or crystals, also act on each other 
when in close proximity. There arises, besides the 
endoleptonic field of force effective in the individual 
structures, an interstitial field depending on the 
reciprocal relations of the component substances, and 
which, therefore, is not constant for each type of 
atom, but is a function of the nature and arrange- 
ment of the neighbouring ones. What is observed 
in chemical processes is a consequence of this inter- 
action, whether it be a change in arrangement, asso- 
ciation of previously separated similar or dissimilar 
particles to higher units, disruption or substitution, 
or whether the action be partly more physical or 
definitely chemical. The setting up of a physical 
field, a change of temperature in particular, or a 
change of pressure, may initiate similar processes 
or modify them. In other words, all physical and 
chemical actions of substances proceed in accordance 
with these ideas. 


VI. THE SERIES OF TRANSFORMATIONS OF 
MATTER 


See OULDS LIQUID CRYSTALS) CRYSTATS 


HE broadest survey of the general physical 
relations of fine-structural aggregates under 


the influence of attractive and repulsive forces 
anisotropically directed, is afforded by a considera- 
tion of the changes of state which all substances pass 
through when their physical conditions are altered. 
By changing the temperature of a substance, that is, 
by speeding-up or retarding the motions of its fine- 
structural particles, it is possible, as is well known, to 
pass the substance through a long series of meta- 
morphoses extending through the gaseous, liquid, 
and solid crystalline states. Thus H,O, for example, 
traverses the states, water vapour = liquid = ice, 
and on the basis of the mechanical theory of heat, 
we are in a position to form a very clear picture of 
these changes. 

In the gaseous state there is very considerable 
leptonic unrest. With individual motions of the 
velocity of a bullet the particles speed hither and 
thither, although only travelling minute distances 
in straight paths; colliding, rebounding, thrusting 
aside, each molecule wins for itself a portion of space 
the size of which is the same for all gases. That the 
same number of material particles of any type take 


up the same space is the essence of Avogadro’s 
61 


62 CRYSTALS AND MATTER 


Hypothesis. J. Loschmidt (1865) was able, in 
furtherance of this hypothesis, to estimate the num- 
ber of particles per c.c. Ato° C. and one atmosphere 
pressure there are 27:6 trillion ; truly a dense popu- 
lation of space, although, however, it should again 
be emphasised that from particle to particle a space 
of average dimensions thirty to forty times the size of 
one particle must be assumed. The inter-connection 
of the individual atoms or molecules by force fields . 
is therefore very slight in gases; the resistance to 
mechanical subdivision of masses of gas, in a certain 
sense the hardness, is correspondingly small. More- 
over, the disperseness depends on the pressure in 
accordance with Mariotte’s law, and on the tempera- 
ture, an alteration of 1° C. involving a change in 
volume of = for any gas whatever. A permanent 
arrangement of the particles, say about some instan- 
taneous centre of their motion, is prevented by their 
rapid movement and diffuse distribution, for they 
roam about passing from one place to another by 
irregular diffusion. 

In liquids, on account of the smaller distances 
of the particles, a field loosely binding the molecules 
is present, as well as the endoleptonic forces, and 
with this comes increased resistance of the mass 
to subdivision, as the viscosity indicates. Indeed, 
internal friction may often increase to considerable 
hardness, as, for example, in glass, which is to be 
regarded in a physical chemical sense as a “ rigid 
liquid.’”’ Silica glass in its cohesion stands only a 
little way behind quartz. For all liquids, the hard- 
ness varies with the temperature. Warm water 
flows through a funnel much faster than cold, owing 
to there being a large diminution of the internal 


TRANSFORMATIONS OF MATTER — 63 


cohesion of the particles, while glass on heating ap- 
proximates to an ordinary liquid. 

In general, a permanent arrangement of the par- 
ticles does not arise in liquids. The case is different 
for the so-called liquid crystals of O. Lehmann. 
There the molecules arrange themselves in the inter- 
stitial field of force more or less regularly oriented 
with respect to each other, very often with one 
direction parallel, so that the anisotropy inherent in 
each one is shown by the single optical axis (as in 
a crystal with a simple whirl axis). Three-dimen- 
sional periodicity of arrangement, however, is lacking 
in this microcosm. Thus Huckel, on investigating 
such substances with X-rays by the Debye-Scherrer 
method, did not obtain crystal diffraction patterns. 
On the contrary, there appeared only the indistinct 
interference ring shown by amorphous _bodies.} 
These so-called liquid crystals are not then strictly 
crystals, but rather intermediate stages to true 
crystals, and as such “ penecrystals,’’ as one might 
call them, they are of very great interest in science. 
Their discovery and preparation by O. Lehmann and 
D. Vorlander, in particular, is one of the finest 
achievements of science. 

Still further connections and transitional stages 
between the structures of individual leptons and 
typical crystals would possibly, if not probably, 
be established if large molecules containing many 
atoms, such as those of albumen and starch, with 


1 In spite of this result, it would be of interest to investigate the 
effect of passing the rays perpendicular to the common direction of 
the molecules. Quite possibly interference diagrams would be 
obtained of the type shown by fibres and flakes, although less 
distinctly than for these substances. 


64. CRYSTALS AND MATTER 


numerous similar groups in their structure,! were 
capable of existing in space-lattice arrangement 
wholly or in part. They would then give a Debye- 
Scherrer crystal effect, contrasting, possibly only 


Fic. 84. 
Fics. 82-84.—Schemes for the gaseous, normal fluid, and liquid crystal states. 


in the matter of degree, with ordinary molecules, 
which, owing to the small number of similar struc- 
tural groups in the kinetic unit, do not furnish a 
space-lattice arrangement. [Let us assign, in this 


1 Probably, too, atoms containing many electrons. 


TRANSFORMATIONS OF MATTER 65 


connection, at least eight equal valued particles 
to an elementary parallelopiped. Substances with 
formule such as Cys9H729Niig5¢Oo43 (Serum-albumen), 
CrssHicosNigsOoisFeS; (Dog’s Hemoglobin), gliadin 
with thirty-eight molecular radicals of glutaminic 
acid, and similar complexes, should possess enough 
similar groups for crystalline structure in the mole- 
cule; aggregates of only about thirty atoms of 
carbon in graphite still show space-lattice character 


Fic. 85.—Scheme for the crystal state. 


according to Debye and Scherrer, as has already 
been mentioned. 

The results of the experiments of P. Scherrer, 
R. O. Hertzog, and W. Jancke on cotton, cellulose, 
starch, etc., for which the crystal effect with X-rays 
was observed, certainly deserve further consideration 
from the above point of view as being evidence for 
the fine-structural nature of “crystalline molecules.”’ 

On crystallisation the already anisotropic particles 
arrange themselves in regular fashion into a space- 
lattice ; by this the formation of external surfaces 
in accordance with Hatiy’s Law (p. 8) is rendered 

5 


66 CRYSTALS AND MATTER 


possible. Thus, in the external ornamentation, we 
have a reliable criterion for the crystalline nature of 
a substance. A fine-structural medley of individual 
leptons gives under the uniform action of the surface 
forces, a spherical form to the lepton complex 
(Fig. 81, p. 59), a form, moreover, not foreign to 
crystalline materials, especially when very small 
masses are considered. Fig. 86, showing drop-shaped 
globulites, slightly curved longulites, pearl-like mar- 
garites, in strings, and hair-like trichites, gives some’ 
elegant examples from the mineral world. With 
larger crystallisations of benzophenone and ice, for 
example, R. Nacken has also obtained circular crystal 
forms by using special methods of cooling. 

The great disperseness of gases is enormously 
diminished on crystallisation; in liquids, too, in 
general, the same thing occurs.!. In this connection 
some figures for sodium and the diamond, as two 
extreme types, will be of interest to the reader. 


Number of atoms per c.c. Ratio. 
1. Gas at boiling-point (820° C.) . ‘ : A 5°5 * Io" x 
2. Liquid at boiling-point . ; - 19,500 x 1038 355 
3. Liquid at solidifying- point, 97: 6° 5 ‘ -') 24,500 *hiGr mn 
4. Crystal at solidifying-point 2 ‘ : . 25,000 x 10!8 f 13 


In solid sodium, with its body-centred space- 
lattice (only two atoms in the elementary cell) and 
its large a-value for the cube edge (4:3 x 10> 8 cm.), 
we are dealing with a soft metal. The condensation 
from gas to fluid is considerable; that for fluid to 
crystal, small. 

With the hard, compact diamond (eight atoms in 
the elementary cell, 4 = 3-53 x I0~ ° cm); aaimuen 

1Exceptions to this, as in the case of H,O, for example, are 


explained by assuming molecular variations in the liquid when the 
crystallisation point is reached. 


Fic. 86.—Globular and curvilinear crystals of microscopic 
dimensions 


is A, =! ie 
¢ i a ¢ i 
ek, y ha 
te ? 
' ; are ; 
a 7 4 
ey ; 
1 bn 
) 
7 wt 
iy 
| 
f ; 
' . 
4 , 
eh 
4 ‘] 
We TRE LIBRARY 
; 7 at 9 Lee 
: Wr een 
~~ gurbeagert OF TRLIRBES 
: . 
j . 
) 
i 
} e | 
| ‘ ’ 
! @ 
; 
: : 
ya! 
ii a ee es 
: . | F, 7 x in’ ie 
= ; ce ake fe ‘ 
Is me ; io, “aN te 
K, at 34 ‘ c ue 
eee ae 
ra ea! aie : ee 
ON ae 


TRANSFORMATIONS OF MATTER 67 


more extensive condensation occurs on crystallisa- 
tion; I c.c. of this gem contains 180-000 trillion: 
carbon atoms, compared with 1-3 trillion in carbon 
vapour at 5500° C. Naturally, with such close 
packing of the particles of matter, the ree be- 
tween them becomes enormous. 

A contraction of the substance in crystallisation 
until the external surfaces of the atoms touch is not, 
however, to be assumed, in view of the known possi- 
bilities of diffusion in the crystal. With change of 
temperature zeolites absorb and expel H,O through 
their siliceous substance, liquids such as carbon 
disulphide can pass in and out of dehydrated chaba- 
site, gold atoms penetrate lead to a perceptible extent 
in a short time, and irregular structures in isomor- 
phous mixtures of metals or salts adjust themselves 
on tempering by wandering of the particles in the 
solid crystal. The atoms must, therefore, in these 
cases be “ able to pass by one another.’’ The con- 
ception of “‘ close packing of spheres,’’ so useful in 
crystallography, must not therefore be understood 
as an actual contiguity of the material atoms. As 
has already been stated on page 46, the matter here 
is one of a subdivision of space, dependent to some 
extent on the external conditions, into “spheres of 
influence ’’ around the atoms. 

In correspondence with these relations the kine- 
matic conditions alter as the series of metamorphoses 
of matter is passed through. While the motion of 
the individual leptons in the ideal gas state of 
matter, neglecting collisions of the particles, is prac- 
tically independent of the surroundings, this free- 
dom of path is restricted in liquids by the reciprocal 
force fields, and the liberty of the structural units 


68 CRYSTALS AND MATTER 


of crystals amounts merely to a tenth of the distance 
between the atoms. Lower temperatures naturally 
signify here a slowing down and limitation of the 
motion of the structural groups, and we may imagine 
that at very low temperatures the particles in a 
certain sense ‘freeze hard.’’ But even at the 
absolute zero the energy of intra-atomic motion still 
remains ; the frictionless agitation of the structural 
units of the atom determines the general constitution 
of matter, and isyeternal. 

The types of fine-structural state are therefore 
easily distinguished. Differences in the motion, in 
the distances apart, and in the mutual interaction of 
the particles, determine the constitution of these 
states under discussion, and thus nothing is more 
natural than that changes of state should modify 
the structure of the particles. If this does not occur 
to the extent of an actual change in the chemical 
character of the substance on transition from the 
gaseous to the fluid and solid states, the identity of 
the molecules in the different states of aggregation 
cannot, after what has been said, be admitted: the 
particles of gases and liquids are changed in passing 
through the series of metamorphoses of matter,} 

Further, in the opinion of physical chemists, the 
state of affairs is complicated still more in the gaseous 


1In this connection the physical condition of a substance is 
occasionally indicated in the abbreviated symbol for its chemical 
nature as a printed formula. Just as the charges of the ions are 
expressed by * and ’, or + and —, the gaseous, liquid, amorphous, 
and solid crystalline states are denoted by the symbols }, ~, - +, and 
—— (the usual crystallographic sign) respectively. For example: 


H,0, H,O, H,O; and. SjO,. for silica glass. In chemical equations, 


aT, oo 
too, such as MgCO, = MgO + CO,, the actual physical condition of 
the substances is shown immediately. 


TRANSFORMATIONS OF MATTER 69 


and fluid phases by equilibria between different types 
of molecule. 

A. Smits follows up this idea in his theory of 
polymorphism, even to the extent of relating the 
process of crystallisation of substances without 
decomposition to an internal equilibrium between 
different types of molecule. 


DISCONTINUITIES OF LOWER ORDER IN THE TRANS- 
FORMATION SERIES. POLYMORPHISM. ENANTIO- 
MORPHY. 


Besides the reverse changes from the gaseous to 
the liquid state,1 and from these to the crystalline, 


Series of metamorphoses of matter 


Total energy 


Temperature decreasing ———» ~973° 


Fic.£87. —Subdivision of the series of metamorphoses. 


within each of these states (and also for liquid 
crystals) less drastic discontinuous transitions are 
possible. The relations of crystals whose greater 
capacity for changes of internal structure is obvious 
in their frequent polymorphism? are here again 
typical. 

1 A transition which can be accomplished continuously by appli- 


cation of a certain pressure and temperature. 
2 Termed allotropy for elements. 


70 CRYSTALS AND MATTER 


Examination characterises these crystal modifi- 
cations as forms of the same substances in a chemical 
sense, which differ from each other in their energy 
content. In the fine-structure this is expressed as 
a variation in the arrangement of the component 
particles, that is, in the architecture ; the stabilities 
are different, and tend, therefore, to cause trans- 
formation of one modification into another. Transi- 
tion occurs if the difference is so considerable that 
the internal resistance to structural change can be 
overcome. 

This may be brought about by a correspondingly 
large change in the external conditions, generally 
the temperature or pressure, and sometimes both ; 
from among many cases of this we mention, as 
instructive examples, the abrupt change of borazite } 
and of a-quartz= f-quartz at 575° (Figs. 88 and 
89), or the transitions of ice. According to a diagram 
worked out by G. Tammann, ice passes through five 
different forms with increase of pressure at — 30° C. 
Occasionally the catalytic influence of a chemical 
field, that is, the intimate proximity of a certain 
substance, brings about a transition which would not 


1 The investigation of this mineral gives very elegant physical 
chemical demonstrations of fine-structure if a borazite plate is viewed 
between crossed nicols and its temperature raised. Owing to the 
considerable double refraction it transmits bright polarisation colours. 
On exceeding 265° the isotropy of the a-borazite spreads out from a 
point like a dark curtain over the previously radiant field, fluctuating 
with each small variation of temperature, and finally enveloping the 
whole in deep shadow. On cooling, the curtain rolls back and the 
structural particles are restored to their old rhombic equilibrium 
arrangement. In other cases change of modification is immediately 
evidenced optically by a colour change. Red mercuric iodide is a 
striking example of this. On heating above 126° it changes from 
its tetragonal equilibrium arrangement to a yellow rhombic form. 


TRANSFORMATIONS OF MATTER 71 


otherwise occur. H. E. Boeke and the author have 
shown that the iron sulphide of magnetic pyrites 
serves aS an example of this; transformation only 


Hexagonal Tsometrical 
a-Quartz a-borazite 
575° ————_—__—__ | 265° 
Trigonal Orthotrimetrical 


(rhombical) 
Cy 


B-Quartz 


Fic. 88.—Homoomeric modification of quartz and borazite. 


Fic. 89.—a@ and 4. Laue diagrams of 8-quartz and a-quartz. After F. Rinne. 


takes place in the presence of a little carbon or excess 
of iron. 

After a change in the fine-structure a new equili- 
brium arrangement is in every case established, 


72 CRYSTALS AND MATTER 


which need not, however, represent the most stable 
configuration under the existing circumstances, but, 
on the contrary, may be, in accordance with Ostwald’s 
step-rule, that which in energy content is nearest the 
original state. 

For polytype modifications such as those of car- 
borundum, discovered by H. Baumhauer, H. Espig’s 
experiments, carried out in my institute, have shown 
that equally large elementary cells contain the same 
number of molecules (24). 

These modifications cannot differ, therefore, in 
specific weight; they will, nevertheless, require 
different amounts of energy to change their molecular 


cei <a > 
7 ee NEES vo ee 


Carborundum Carborundum Carborunduin 
Type I Type IT Type LIT 


Trigonal Hexagonal 


Fic. 90.—Polytype modifications of carborundum. 


motions, i.e. their specific heats are different. <A 
change of one carborundum modification into another 
is not known. Such changes of structure are to be 
observed, however, in other cases, for generally the 
modifications are structures with definitely unequal 
stabilities, except at the transition point. 

If the difference in the fine-structure is not too 
great the modifications are homdomeric, that is to 
say, transformable one into the other without break- 
ing up the whole structure. This is the case for the 
quartz referred to, and also for borazite and leucite. 
With these minerals it is easy, by heating one modifi- 
cation above the limiting temperature to change it 
over into the other, and, on reversing the process, 


TRANSFORMATIONS OF MATTER 73 


by lowering the temperature to return to the first 
modification as often as desired. The solidity of 
the crystal structure does not preclude exact ex- 
periment, and Laue diagrams give a picture of 
this “dance of the leptons’’ in the change from 
B- to a-quartz. In this case the ternary arrange- 
ment of the particles becomes, as though at some 
mysterious word of command, a senary arrangement, 
as Figs. 89a and 0b show, a truly wonderful glimpse 
of a fine-structural sanctuary of nature. A similar 
case arises with iron, and here the investigations of 
A. Westgren have disclosed the relations of the fine- 
structure. Four modifications 


of iron are known—a, f, y, | 
and 6 irons. On heating to | 
above 769° the a passes into 


Ppomemvanicry, at -oo6" this "0 ¢ 
gives the Y modification, and Fic, g1.--- Stereograms of iron 
finally an increase of tempera- aie eet So a 
ture to 1401° leaves the metal in the 8 state; the 
transitions in every case are indicated by thermal 
changes. 

The transformation a= 8 is very clearly shown. 
Here at 769° lies the upper limit of the technically 
important property of ferromagnetism (capacity for in- 
tense magnetisation) possessed by the metal. In their 
fine-structure a and 8 irons are closely allied, differing, 
however, from the y metal; Westgren was able to 
_establish that the former have body-centred lattices, 
whilst that of y-iron is surface-centred (Fig. 91). 
The transition from one variety of iron into another 
takes place quite smoothly. On the other hand, 
with carbon tetrabromide for example (Fig. 93), 
clearly marked fractures arise on slowly heating or 


74 CRYSTALS AND MATTER 


cooling, and even more so in the transition from 
diamond to graphite, an example of allomerism 
which P. Debye and P. Scherrer have made classical. 


Planes of the cube, rhombic dodecahedron, a pyramidal 


Planes of the cube, rhombic dodecahedron, a pyramidal cube, 


cube and the octahedron. 
and the octahedron. 


Fic. 92a.—Atom domains of a-iron.! 
Fic. 924.—Atom domains of y-iron. 


In diamond the carbon particles are arranged in 
aliphatic tetrahedral grouping, but in graphite they 
are arranged in the aromatic ring form which is 


1 The lower scales in Figs. 92a and 6 relate to depth distances. 


TRANSFORMATIONS OF MATTER 75 


a very different arrangement, and one requiring 
much more space from ring to ring. Graphite is 
in a certain sense an elongated diamond (Fig. 94). 


Fic. 93.—Change of modification of carbon tetrabromide, the external form 
remaining intact but showing internal mosaic formations. 


The restoration of the atoms to the diamond arrange- 
ment has not yet been accomplished. 
In conclusion, the remarkable enantiomorphy of 


Diamond © 
a-250-10-8com 
€=306-10-8om c -5 97: 10-5an 
@Ce7123 aca F203 


Fic. 94.—Allomeric modifications of carbon (diamond and graphite). 


crystals, as shown by quartz and cane sugar, for 
example, should also be mentioned. 

The two varieties which call for consideration 
are differentiated as right- and left-handed; as 


76 CRYSTALS AND MATTER 


regards their macro- and micro-stereochemistry, each 
variety taken by itself is without mirror symmetry, 
but the two together are symmetrical one to the 
other, as the halves of the human body. Their 
stability to changes of external conditions is exactly 
the same. They are obtained mixed together indis- 
criminately, and are not convertible one into the 
other. 

Clearly then the modifications of a crystalline 
material like the isomers of liquids and gases are 
chemically different substances, although, at the 
same time, the divergences in their material relations 


, ly right left 
Cane sugar 


Fic. 95.—Enantiomorphy of cane sugar. 


are, for the most part, not very large. In the case 
of carbon, however, they are not small (graphite is 
oxidisable to graphitic acid, diamond is not), nor 
with calcium carbonate are they inconsiderable 
(boiling with cobalt nitrate solution colours aragonite 
violet, but calc-spar blue) ; quartz, finally, scarcely 
shows them at all. 


REVIEW AND UNIFIED CONCEPTION OF THE 
STATES OF MATTER 
A survey of the whole series of metamorphoses 
from the gas to the last modification in the tempera- 
ture scale presents a picture of discontinuous energy 
changes, and between these discontinuous changes 


TRANSFORMATIONS OF MATTER 77 


steady variations within each state, which possibly 
represent energy exchanges in minute jumps. Ac- 
cording to that idea, large energy quanta are com- 
posed of small quanta, which for the observed 
continuous changes follow each other in a steady 
series, whilst for the sudden discontinuous changes 
they rush over a steep fall. Structurally, in the 
former case one is concerned with fine leptonic jumps, 
in the latter with motions of larger leptonic complexes. 
Regarding the fine-structural changes of sub- 
stances in this way, we arrive at a unified conception 
of the states in which matter is presented to us. 
Starting from the absolute zero and continuing to 
the highest temperature obtainable, matter passes 
through a series in which the transition points 
(whether these be boundaries between the solid, 
crystalline, liquid, or gaseous states, or terminating 
points of sub-states within these) represent obstacles 
on a constantly rising road, which must be overcome 
by especially large expenditure of energy. The 
substance deviates more and more from the character 
it possesses by virtue of its own particular constitu- 
tion. In ordinary experiments carried out at room 
temperature, that is, at about 290° absolute, we must 
in consequence assume considerable restrictive in- 
fluences at work on the crystal, resulting in modifica- 
tions of its proper constitution.!. The investigation of 
matter at very low temperatures eliminates, as much 
as possible, the enormous influence of the surround- 
ing circumstances on the primitive structure. Thus, 
of all the states of aggregation, the solid crystalline 
approximates most nearly to that of ideal matter. 


1 Of these, temperature is the most active; after that, pressure 
and chemical action are the most important. 


VII. GENERAL TECTONIC ARRANGEMENTS 
IN THE FINE-STRUCTURE OF CRYS TAS 


ANALYSIS OF STEREOCHEMICAL FORMULA 


structural researches from a chemical stand- 
point suggests many questions of general 
importance. 

First, it is an obvious conclusion that the sug- , 
gested schemes express the analytical constitution 
of the substances they represent. It is, of course, 
necessary to argue correctly from the fine-structure. 
For example, the diagram for the stereochemical 
formula of fluor-spar CaF, (Fig. 96a) contains in the 
tectonic unit there depicted 14Ca and 8F atoms, 
leading apparently to the incorrect formula Ca,,F, ; 
but it must be remembered that each Ca at a corner 
of the figure is shared by eight cubes, since the 
fine-structure is repeated indefinitely, often on all 
sides. Consequently, each of these only represents 
an eighth of a Ca. Similarly, a Ca on a surface of 
the cube (as on a wall separating two cubes) must be 
counted as $Ca, whilst the 8F atoms in the inside 
of the figure must be reckoned in full. We have, 
therefore, (8.4.+6.4)Ca and 8F, or Ca,F, corre- 
sponding to the actual CaF,. In the same way, the 
reader will easily decipher the stereogram of iron 
carbide Fe,C, and the formula of ice H,O (Figs. 
96d and c). 


A CONSIDERATION of the results of fine- 


78 


TECTONIC ARRANGEMENTS 79 


If, according to this, a stoichiometrical balance 
for a portion of the crystalline body surrounded by 
others is recognised, it seems unmistakable that the 
crystal as a whole does not conform to the require- 
ments of the law of simple multiple proportions, if 
its space-lattice structure extends to the external 
surfaces. In this case the particles lacking neigh- 
bours would not balance analytically, and there would 
be an excess of one kind. The edge of the elementary 
cell of fluor-spar (shown in Fig. 96a) is 5:44 x 1078 
cm.long. A cube of the mineral of 5-44 x I07? cm. 


Fic. 96.—Stereochemical formule. a. Fluor-spar CaF,. After W. H. and 
W. L. Bragg. 6. Iron carbide Fe,C. After A. Westgren. c¢. Ice H,O. 
After G. Aminoff. 

side which is thus of colloidal dimensions would 
give, in accordance with the above scheme, the ratio 
Ca: F = 36:51 : 63:49, instead of the theoretical value 
33°33 : 66:66. By increasing the size of the cube the 
deviation from the ideal ratio would gradually become 
smaller, and, finally, analytically indetectable. Since, 
however, it is unlikely that for crystals the law of 
simple multiple proportions does not hold good, there 
only remains the assumption that the regular space- 
lattice arrangement of the particles fails in the 
surface zone. 


80 CRYSTALS AND MATTER 


VALENCY 


No less important chemically is the question as 
to how valency is exhibited in crystals. We may 
expect, therefore, that the study of the specially 
regular crystalline materials will lead to a better 
understanding of the controversial nature of valency. 
For gases and liquids the only rules of valency 
which have actually been experimentally confirmed — 
are restricted mainly to the specification of how 

many atoms have found a place 

round another atom under de- 

finite chemical and physical 

conditions. With regard to the 

fine-structure, it is of interest 

to note that, in substances with 

simple valency, it is rare that 

ZE. more than eight atoms are united 

to one of another (as the series 

FNa; F,Mg; B,Al; 2, Sip eye 

B.S; —; F,0Os shows), and that 

these combining numbers for a 

Fic. 97.—Net ofasimple given atom depend on the tem- 

crystal and its twin form. erature, ie. on the motion of the 

particles. Both of these are clearly structural 
characteristics permitting of obvious explanation. 

As to the spatial arrangement of the particles, 
no definite result is arrived at in terms of valency, 
although it appears quite natural in CH, for us to 
place the 4H satellites round the central carbon atom 
in similar positions on the corners or sides of a 
tetrahedron (compare Fig. 56, p. 43). In particular, 
it is entirely a formal characterisation if valency or 
co-ordination bonds are represented as single directed 
lines of force between the atoms and molecules. In 


TECTONIC ARRANGEMENTS 81 


Fig. 97 it is clearly seen that such bonds between 
similar atoms at the common surface of a twin have 
different directions from those in the simple crystal. 
We must, however, note that such tensors have only 
the character of pedagogical simplification, and not 
that of reality, just as the customary representation 
of chemical formule on paper is, to a certain extent, | 
merely a simplified projection of very complicated 
spacial arrangements. The “structural chemical ”’ 
mode of representation in a plane, and the drawing 
in of directed bonds, are but symbolisations, super- 
ficial and linear respectively. Plane arrangements 
of the atoms or of their centres of motion are not, 
of course, excluded; for benzene, in fact, such an 
arrangement is very probable. In this example we 
have a limiting case in which matter is practically 
reduced to a plane. A methylation of benzene leads 
at once to a definitely three-dimensional molecular 
structure.t. In general terms, the constellation of the 
particles must be thought of as a complex action of 
the anisotropic fields of force, gradually weakening, 
which always surround the particles. As is well 
known, especially in inorganic chemistry, we cannot 
get on without introducing some spacial co-ordination 
action besides valency ; A. Werner has discussed this 
very ingeniously. Organic chemistry, owing to the 
special nature of the nucleus of all its compounds, 
carbon, for which the valency and co-ordination 

1 This does not lessen the great value in teaching the notions 
of directed singular valency and co-ordination forces, and their 
representation as lines in plane formule. It is an indication of the 
considerable practical utility of this handy method of representation 
that such a formulation of the certainly very complicated interaction 
of the oscillating particles generally guides us safely through the 


innumerable phenomena of chemistry. 
6 


’ 


89 CRYSTALS AND MATTER 


numbers are the same, might easily attempt to dis- 
pense with such conceptions. 

In contrast to these uncertainties in the case of 
amorphous materials, X-ray experiments on crystals 
have disclosed not only how many particles at any 
time surround one atom, but, in addition, details as to 
the spacial arrangement of the centres of the particles 
in terms of centimetre co-ordinates. Thus it is 
known definitely that, in the case of diamond, each 
| C atom is placed at the 
centre of a tetrahedron, 
at the corners of which 
are four others, each 
1°53 X 10~° cm. from the 
centre; in rock-salt each 
Na atom is surrounded at 
a distance of 2:81 x 1078 
cm. by 6 Cl atoms on the 
normals to the surface of 
a cube, and each Cl simi- 
larly by 6 Na’s.1. In addi- 
tion, spacial representation of the atomic domains 
gives us a Clear insight into the dynamical relations ; 
in Fig. 98 this is done for sodium chloride. 

All the other rules of valency lead us here to 
somewhat insecure ground. It is true that in crystal 
stereochemical figures connecting links may be looked 
upon as valency tensors, and it is in this way seen 
that in the diamond the tetravalency of carbon is 
clearly shown graphically. In other cases, a val- 
ency distribution is symbolised, as for rock-salt in 
i values, and in similar ways for other substances. It 
must not be forgotten, however, that the methods 


Fic. 98.—Axial arrangement of the 
atom domains in rock-salt. 


‘See also Fig. 96, p. 79, and Fig. 24d, p. 22. 


TECTONIC ARRANGEMENTS 83 


discussed above are only graphical aids, and merely 
summarise diagrammatically the field of force. 

In this connection it must be ascertained whether 
any further knowledge of valency relations is available 
from a consideration of the crystallographic evidence 
of a definite “form ’’ for the structural particles : 
and here we are helped by ideas of H. J. van’t Hoff, 
who assumed that it is the definite form of the atoms 
which involves directional maxima in the tendency 
to aggregate. These maxima therefore determine 
the fine-structural distribution possible, and therefore 
the valency. 


Fic. 99.—Valency schemes for a and 4 Rock-salt ; ¢, calc-spar. 


Thus a carbon atom, owing to the tetrahedral 
arrangement of its external electrons, presents a 
specially stable distribution for four hydrogen atoms 
at the corners of a tetrahedron rotated with respect 
to the carbon tetrahedron, so that the H atoms lie 
over its four faces (Fig. 56, p. 43). The normal 
quadrivalency of carbon arises then primarily as the 
result of a stable arrangement in the field of force, 
and kinetic relations are therefore of importance here. 

This anisotropy in the shape of the atom will exist 
also in the atomic domain, concerning which refer- 
ence is made on page 46; in these domains, however, 
the corners and edges will be rounded off. 


84. CRYSTALS AND MATTER 


A zero valency arises if a stability already high 
permits of no further additions. The relations of 
substances possessing the character of rare gases 
may, following the ideas of Bohr, W. Kossel, and in 
particular H. Tertsch, be regarded in this way. 
After filling an electron zone with 8, again 8, then 18 
(= 6+ 12), again 18, and finally 32 (= 24 + 8) 
electrons corresponding to the periods in the system 
of the elements,’ a shield so stable from outside in- 
fluence is constructed that a fine-structural deforma- ° 
tion and chemical change is rendered difficult to a 
high degree. Thus zero valency is obtained in each 
case. Figs. 100-103 represent these stable electronic 
arrangements diagrammatically in crystallographic 
form. Helium (with two negative electrons) has 
probably pinacoidal stability (Fig. 30, p. 29).2 The 
other numbers, 8, 1% =6+12, 32 =8-4+ 24, are in 
accordance with the crystallographic motives of the 
cube (six surfaces, eight corners), the rhombic dode- 
cahedron (twelve surfaces over the edges of the cube), 
and the pyramidal cube (twenty-four surfaces over 
the six of the cube). 

In the construction of molecules analogous cases 
are not improbable. That tetrahedral grouping, as 

17 H, 2 He; 1 Li, 2 Be, 3 B, 4.C, 5'N, 6 O, 7 F, 3 Nee 
2 Mg, 3 Al; 4 Si, 5 P,'6'S;4 Cl, 8 Ar; 2K, 2 Ca, 3'8cua eee 
6.Cr, 7 Mn; 8 Fe, 9 Co, to Ni, 11 -Cu, 12° Zn, 13 Gay 4) ee 
16 Se, 17 Br,.18 Kr; 1 Rb, 2 Sr, 3-Y, 4 21, 55ND) 0))iGw eee 
9 Rh, 10. Pd, 11 Ag, 12 Cd, 13 In, 14 Sn, 15 Sb, 16 Te, 170} p ee 
t Cs, 2 Ba, 3-19 ‘the rare earths, 20 W, 21-22 Os,'23 If, 24° Pte aera 
26 Hg, 27 Tl, 28 Pb, 29 Bi, 30 Po, 31-32 Em; t-2 Ra, 3 Ace 
5 Br, 6 U, 

* The total electrons in the negative shell may probably be 
arranged for the rare gases in the following way: He2; Neonz2 + 8; 
Ar 2+8+8; Kr 2+8+6+12+8; X=2+8+46412 
+6+12+8; Em=2+84+8+6+412+6+4 12 + 244 8. 


TECTONIC ARRANGEMENTS 85 


Fia. 100. Fic, 101. 


Fic, 103. 


Fics. 100-103.—Crystallographic schemes for zero valency due to perstable 
electron arrangements and uranium as the final term in the atom series. 
Rare gases: Shell with 8 electrons for neon and argon; shell with 6 + 12= 18 
electrons for krypton and xenon; shell with 24 + 8 = 32 electrons for 
emanium ; uranium as the final member of the electron aggregates. 


86 CRYSTALS AND MATTER 


is to be assumed for CH,, with its great regularity, 
favours to a certain extent chemical stability is shown 
definitely by the properties of methane. 

To substances which require only small variations 
in their external electron shells to attain the external 
structural balance of a rare gas, we must, on the other 
hand, readily assign high chemical affinity. The 
left and right-hand neighbours of the rare gases in 
the systems of the elements are of this type; the 
alkalies and halogens. Their tendency to combine 
one with the other is great, and according to 
W. Kossel, this is brought about through the loss of 
an electron by the alkali and by a gain of one by the 
halogen. 

The following small table of the names and 
numbers of electrons in the atoms renders this clear: } 


Halogens. Rare Gases. Alkalies. 
I 2 4 
Hydrogen. Helium Lithium. 
9 IO II 
Fluorine. Neon. Sodium. 
17 18 19 
Chlorine. Argon Potassium. 
an 36 ou 
Bromine. Krypton Rubidium. 
ey 54 55 
Iodine. Xenon. Caesium. 


In a similar manner it will be readily assumed 
that molecules, too, with the great internal tension 
which must arise if the stability is not fully estab- 
lished, tend to like transformations. 

Moreover, in connection with these assumptions 
as to the mode of action of valency, the occasionally 


1In view of the analogy between Li H and the halides, H is 
included in this comparison. 


TECTONIC ARRANGEMENTS 87 


observable changes of valency in changes of crystal 
modification, i.e. in stereochemical structural rear- 
rangements, should be noticed. In the diamond the 
quadrivalency of carbon is expressed as four equally 
large maxima in the arrangement. In graphite 
structure, on the other hand, it is seen that a 4 x I 
is differentiated into a 3 + 1 valency. This change 
in the form of the field of force is, to some extent, the 
first step towards a diminution of valency ; it forms 
an intermediate stage to a tri- or divalency of carbon. 
Since it may well be assumed that, 
in the process of changing from dia- 
mond to graphite, a variation of 
atomic form by rearrangement of 
the electrons takes place (which the 
optical differences of clear diamond 
and black graphite, besides con- 
siderations of crystallographic sym- 
Mereyemindicate),* then it 1s quite 
obvious that a change in the shape 00: (oh viee” 
of the atoms goes hand in hand 

with a change in their method of arrangement. 
The idea that morphological changes themselves 
occur under the influence of alterations in the ma- 
terial surroundings, and of alterations of tempera- 
ture affecting the internal motions, harmonises in 
the best possible way with the experimental result 
that the valency of the substance may depend on the 
kind of matter with which it is in valency connection, 
and also on the temperature. 


MOLECULES IN THE CRYSTAL STRUCTURE 


It will appear of great importance to the chemist 
in dealing with these crystallographic stereochemical 


88 CRYSTALS AND MATTER 


formule to investigate the positions of the mole- 
cules which come under his consideration in the 
chemistry of gases and liquids. Occasionally it 
happens that the molecular scheme appears in the 
elementary cell, as, for instance, those in Fig. 24, 
page 22. That has, however, as little foundation 
as the representation of the macroscopic crystal as a 
molecular unit ; for we are concerned in both cases 
with more or less arbitrary sections from the fine- 
structure. For zinc sulphide, Fig. 105 might just as - 


Fic. 105.—A cube section. Fic. 106.—A rhombic dodeca- 
hedral section. 


Stereochemical formula for zinc-blende ZnS. 


well be proposed as Fig. 106, both of them are con- 
sistent with the ratio Zn:S =1:1. It appears from 
this that the discussion of the appropriate architec- 
tural relations must be attempted in some other way. 

In particular, it must be remembered that the 
chemical properties of a substance are characterised 
above all in the organisation of its general architec- 
ture. As we study an example of architectural art, 
such as a cathedral, not only as an architectural 
unit, but recognising its morphological subdivision, 
so it is an important chemical problem to investigate 
crystals in the same way. The customary method 


TECTONIC ARRANGEMENTS 89 


of writing formule as Hg, Cl’, Na~ Cl’, (NH,)° Cl’, 
eee). , NH(CH,).1) C,H, (COOH) (NH) accords 
with this. In such symbols the chemical indi- 
viduality is emphasised, as well as the appropriate 
tectonic organisation, as exhibited in reactions. 
Hence special efforts must be made to find out the 
inner groupings in the architecture 
of crystals. 

On the other hand, however, it 
must be added that the kinetic 
units in the erection of higher 
chemical complexes, as in the ag- 
gregation of atoms to molecules, 
and so to crystal units, may 
become indistinguishable in the 
fine-structure, or may even be dis- 
solved altogether. This is the case, 
for example, in the formation of a 
molecule of H, from its atoms. 
It cannot be ascertained in the 
scheme shown which of the two 
electrons belonged, originally, to a 
given nucleus. The earlier indi- _ Fics. 107 and 108.— 

: 4 ’ Models of the hydrogen 
vidualities are in such extreme atom and the hydrogen 
cases completely destroyed. passa 

Therefore, we are not to restrict ourselves, as a 
matter of course, to a rigorously uniform representa- 
tion of the fine-structure of the crystal, but must 
expect numerous gradations in view of the great 
abundance of chemical conditions which exist in it. 
Naturally, in formal mathematical representation, an 
arrangement in an atomic space-lattice is always 
possible by joining up the similar and similarly 
situated atoms into a lattice unit. This procedure, 


90 CRYSTALS AND MATTER 


however, invests the crystal structure with practically 
no physical chemical properties. The presentation 
of a purely atomistic structure of the crystal, or its 
comprehensive characterisation with co-ordination 
bonds, is obviously too general. 

Considering the results already obtained in crystal 
leptology with a view to a physical chemical inter- 
pretation, the desired organisation into radicals is, 
in fact, often definitely shown in the architecture. 
The Braggs, as originators of fine-structural schemes, ~ 
have pointed out the dumbbell-like connection of the 
two sulphur atoms in iron pyrites, and, in somewhat 
greater detail, P. Niggli has expressed the opinion 
that the representation of the complete crystal as a 
purely atomistic space-lattice complex certainly may 
always be carried out formally, but that many inter- 
connections between certain atoms forming structural 
groups (as P. Niggli called them) stand out clearly 
on a merely architectural consideration of the 
schemes. These structural groups I have designated 
geometrical radicals or leptyles in adaptation to 
chemical ideas. Finally, groups of a molecular 
nature are occasionally unmistakable in the crystal 
arrangement. Such leptyles occur in iron pyrites 
with its doublet S,, in calc-spar with the ion 
CO;, and in rutile and anatase with the molecular 
TiO, complex. <A fine example of leptyle grouping 
has been investigated by Dickinson (‘‘ Journal of 
the American Chemical Society,” vol. xliv., 1922, 
p. 287). Moreover, for organic substances with ring 
formation, such a fine-structural grouping must be 
assumed according to P. v. Groth, who formerly 
brought into prominence the atomistic crystal struc- 
ture. As regards cyclic chains of atoms so funda- 


TECTONIC ARRANGEMENTS 91 


mental chemically, which can with certainty be 
attributed to the molecules of benzene, naphthalene, 
and other organic substances, it is more than probable 
that these chains persist in crystallisation. Crystal- 
lised organic compounds will often be molecular 


Fic. 109.—Physical chemical system of molecules split up geometrically into 
six different space-lattices. 


aggregates loosely knit into space-lattices. This con- 
ception W. H. Bragg has taken as fundamental in 
his present researches on organic compounds. 

In the same connection considerations of cleavage 
can be brought forward. According to E. Schiebold, 


@ 
lron pyrites 72,5; Rutile 77Q; 


@ Fe oS Cy e7 
Fic. 110.—The leptyles S, in Iron pyrites FeS,, TiO, in rutile, CO, in cale-spar. 


such planes are to be regarded as plane rifts in the 
fine-structure, as it is accepted that these surfaces of 
separation do not cut through any strong chemical 
bond. With graphite this is clearly shown, as 
cleavage along the principal plane in the fine-structure 


92 CRYSTALS AND MATTER 


separates out to some extent packs of ring radicals 
with their 3 + 1 valency by cutting through the 
extenuated connecting links. According to A. John- 
sen, twin cleavage with separate motion of the 
structural groups Ca and CO, demonstrates this 
same relation. We are concerned here with a very 
remarkable property of calc- | 
spar which, on the applica- 
tion of a lateral pressure, 
permits portions of the whole 
structure to slide to a position 
symmetrical with the station- 
ary part (Fig. 112). 


Fic, 111.—Structural groups in 
graphite. 


Oo 2 SY 6 8 10 12 14 1616 20 


Fic. 113.—Analogous selective reflexion in the ultra-red of the hydrated sulphates 
of Mg —- K, Co— K, Zn — K. After Cl. Schaefer and M. Schubert. 


In this so-called simple displacement or twin 
slipping, the Ca atoms on one side, the CO; complexes 
of the fine-structure on the other, shift apart each as 
a complete whole. 

The researches of Cl. Schaefer on the reflexion of 
ultra-red radiation by crystalline salts and the con- 


TECTONIC ARRANGEMENTS 93 


stant effect, there observed, of the groups NO,, SOu,, 
SeO,, CrO,, and H,O, also illustrate the idea under 
discussion, as occasionally in these cases definite 
radicals can be separated out partly by chemical 
treatment of the crystal structure. 

In short, there is no lack of well-supported 
- opinions that the crystal lattices are organised to a 
varying degree in natural complexes, more or less 
simple, up to the molecular type. With this in view, 
A. Reis has classified crystal lattices from a chemical 
standpoint, and has emphasised, following the lead of 
W. Nernst, F. Haber, M. Born, and others, the special 
nature of salts as ionic structures. 

Thus neutral atom lattices, atom ion lattices, 
neutral and charged radical lattices, as well as mole- 
cular lattices, are to be differentiated, to which series 
I should like to add one more, the mixed lattice. 
Calc-spar Ca: CO,;, and zeolites such as heulandite,} 
are examples of mixed lattices. The H,O contained 
in the latter mineral is set so loosely in the fine- 
structure that it evaporates from the crystal on 
the temperature being raised, until an equilibrium 
content is reached. On lowering the temperature 
the H,O is again taken up from the surroundings 
without the remaining structural complex being 
damaged or in any way broken up, as is shown 
by the Laue diagrams. That indicates great struc- 
tural independence of the water particles. They are 


1Indeed, fundamentally all lattices with isotopic substances 
(p. 185) have already a mixed character. The isotopes of a given 
atomic variety, say, Cl,;, Cl,;, and Cl,., are, of course, to be 
thought of as separated in space and replaceable one by another 
(compare Fig. 119, p. 98). 


94 CRYSTALS AND MATTER 


to be considered as regularly arranged structural 
particles only loosely coupled to the silicate space- 
lattice. 

Complete kinetic units in the sense of individual 
leptons in ideal gases are themselves quite ineffective 
in the isodynamostasy of the whole; for the leptyles, 
however, this naturally is not so. Like all structural 
particles, they occur in the crystal in complex 
structural connexion with those above and below. 
The kinetic unit for crystalline materials is the © 
whole crystal. In this sense it replaces as a new 


UTI AS 
Ves 
XX 


WN a 


Pigart4, Fig. 125. 


Fics. 114 and 115.—Schemes for atom and radical lattices. 


compound the individual molecule of gases and 
liquids. Within the crystal, affinity tensors, in the 
sense of valency theory, pass in and out from one 
structural particle to another throughout the whole 
structure ; the crystal is in this respect a leptoblast. 
With reference to the cohesion prevailing through the 
whole system, P. Pfeiffer has made the relations clear 
in the designation ‘‘ molecular linkage.”’ 

The cohesion of the structural groups arising as a 
result of force fields, in general, very strong, prevents 
the disrupture which is possible for liquids and gases 
on the application of mechanical force. This cohesion 


TECTONIC ARRANGEMENTS 95 


enables a crystalline material more or less to with- 
stand fracture, and renders crystals, in consequence, 
correspondingly hard materials. 


ho 


5 
& 
RY 


Yeths, yooh, 


2 


Fic. 116.—Scheme for the molecular lattice of an aromatic compound C,H,Xop 
(compare also Fig. 85, p. 65, and Fig. 109, p. 91). 


The chemical facts alone indicate how the inner 
structure is organised in general, yet surprising 


Fic, 117.—Mixed lattice (calc-spar). 


results sometimes arise. This is shown by a re- 
search in my institute by Lotte Berndt on olivine, 
to which is ascribed the formula of an orthosilicate 


96 CRYSTALS AND MATTER 


Mg,(SiO,). According to these experiments a corre- 
sponding organisation into the structural groups of 
Mg and SiO, leptyles (charged) is to be expected. 
The symmetry of the space grouping proves, however, 
that SiO, leptyles do not arise in the fine-structure. 
On the contrary, the atoms in the olivine must be 


< 


/ 
| 


We 
\ 


SF 
= 


\ 


= 


SS 
Ro 
re 


Fic. 118.—Leptoblastic structure for diamond. 


separated into MgO- and Si0,-groups. Presumably 
this will be shown by reflexion experiments in the 
ultra-red, page 92. More comprehensive experiments 
on silica salts, as to their nature as ortho-, meta-, or 
polysilicates, would be of the greatest interest in 
crystal stereochemistry. 


VIII. ASSOCIATION OF THE FINE-STRUC- 
TURAL PARTICLES IN MIXED CRYSTALS 
AND OUT-GROWTHS ON CRYSTALS 


TRANSITIONS BETWEEN CHEMICAL COMBINATION 
AND PHYSICAL MIXTURE. MIXED CRYSTALS 


dealing with the structure of crystals that they 

permit, to a greater or less extent, indiscriminate 
heterogeneity of their constituents without injury 
to their stability. Such is the case for isomorphous 
mixtures. We find here certain atoms or groups 
of atoms replaced by others in the fine-structure of 
the crystals. It is certain that crystals with isotopic 
constituents (p. 185) must be reckoned as isomor- 
phous mixtures. In NaCl crystals, for example, 
three kinds of chlorine atoms are to be assumed, 
Cl,; (i.e. with the atomic weight 35), Cl,,, and 
perhaps Cl,,. Although these atomic sub-types 
have different masses they do not differ chemically. 
Such isotopic mixed crystals represent the ideal case 
of isomorphous substitution.t In addition, in the 
interchanges of isomorphous mixed crystals, we are 
concerned with substituents quite analogous chemic- 
ally ; in the mixtures NaCl and NaBr, Ba(NO,;), and 
Pb(NO,),, KCl and K(CN), for example, the Cl and 


1 Owing to the uniform intermixture of the chlorine isotopes on 
the earth, one has presumably in rock-salt always 77 per cent. Cl,,, 
and 23 per cent. Cl;,, with traces of Cl,). Theoretically, however, 
any arbitrary mixture is possible. 

" 97 


[< is an especially interesting point to note in 


98 CRYSTALS AND MATTER 


Br, Ba and Pb atoms respectively displace each other 
in the fine-structure. This interchange may also 
occur with chemical radicals such as NH,, which 
replaces K, or CN, which is interchangeable, with 
Br. Similarly, the groups’ SO,, S5éO;,*CrOj@aae 
the members of many other series may be substi- 
tuted one for another. 

With such complex space-lattices, it is not uncom- 
mon for optical anomalies! (or in the isometric sys- 


Fic. 119.—Scheme for the isomorphous Fic. 120.—Scheme for the isomor- 
mixture of NaCl,,, NaCls,, NaClyo. phous mixture of NaCl and NaBr. 


tem double refraction in the place of the isotropy 
possessed by the pure components) to indicate, accord- 
ing to G. Tammann, unequal apportionment of the 
substituents. For barium lead nitrate the abnormal 
phenomena disappear with time, even at ordinary 
temperatures or on heating; they are not found, 
therefore, in mixtures which, like pyrop in eruptive 
rocks, crystallised at higher temperature. 


PHYSICAL CHEMICAL SIGNIFICANCE OF THE MIXED 
CRYSTAL 

As regards the X-ray effect, the fine-structural 
heterogeneity in question is not evidenced experi- 
mentally in diffraction, which is an indication that in 
isomorphous mixtures the substitution is, as assumed 
above, complete. If in mixed crystals there existed 
intergrowing layers of first one and then the other 


+R. Erauns has given from the results of very many actual 
observations an excellent summary of these anomalies. 


FINE-STRUCTURAL PARTICLES — 99 


substance, such as very thin sheets of KBr and KCl 
growing together to give the mixed crystal (KBr 
and KCl), then each component would naturally give 
its own interference pattern in the Debye-Scherrer dia- 
gram. This, however, is not the case. On the other 
hand, Hadding found such double patterns for Nak 
felspar, showing that here one has a sub-microscopic 
mechanical mixture of the two felspar varieties, and 
nota simple interchange of sodium and potassium ions. 

How are we, therefore, to think of this newly im- 
planted portion of the fine-structure in combination 
with the whole? These matters lead to the con- 
sideration of a fundamental question in chemistry, 
namely, the relation between chemical compound 
and physical mixture. 

If the existence of mutual coupling by the action 
of forces effective only very near the atom is to be 
emphasised, then isomorphous mixtures must come 
under the headings of chemical compounds. All 
atoms in a mixed crystal stand certainly in definite 
chemical connexion one with another. Moreover, no 
reliable line of demarcation between compounds and 
mixtures can be laid down in passing from one pure 
crystal through the series of mixtures to the other 
pure substance. In the unmixed calcium carbonate 
of calc-spar, for example, a chemical compound in 
every sense exists. If now a manganese atom occu- 
pies one of the numerous corresponding places not 
occupied by a calcium atom, then certainly the 
constitution of the whole as a chemical compound 
is not abruptly destroyed. Thus, by repeated sub- 
stitution of atom for atom by correspondingly small 
quanta in the crystal, one may pass right to the other 
end of the series CaCO; — MnCOQ,. 


100 CRYSTALS AND MATTER 


It is impossible to draw a line between chemical 
combination and mixture. The case of alteration in 
the structure of zeolithic isomorphous mixtures will 
be similarly judged. In these silicates more and 
more Ca, for example, can be replaced by Na, if 
a solution of NaCl is allowed to act on a crystal con- 
taining calcium. In accordance with the laws of 
mass action, a crystallographic chemical change in 
the structure takes place, which presents in its 
various stages a complete series of isomorphous 
mixtures from the Ca to the Na, mineral. In con- 
sequence, the structural parts must be in chemical 
combination with one another. 

Although then isomorphous mixtures in the 
regular chemical coupling of their structural parts 
suggest chemical compounds, still on the other hand 
they do not accord with the second characteristic of 
such compounds. As H. Kopp has already empha- 
sized, the distinguishing feature of a chemical com- 
pound is constancy of atomic proportions, in spite of 
very great variation of the original conditions. 

Typical molar mixtures, the constitution of which 
follows change of the external conditions in a con- 
tinuous fashion, show the opposite effect. Evidently 
this is the case for isomorphous mixtures. If, for 
example, the KBr content of a mixed solution of 
KBr and KCl be steadily increased, the composition 
of the mixed crystals (K(Cl, Br)) separating varies 
continuously in correspondence. 

Meanwhile we recognize immediately, in these 
isomorphous mixtures, that the difference in ques- 
tion is bridged over in the fact that the continuous 
variations of their chemical constitution are actually 
at the same time discontinuous. It is only because 


FINE-STRUCTURAL PARTICLES | 101 


of the atomistic, and therefore analytically very 
small, discontinuous changes in that large leptonic 
unit, the crystal, that practical continuity obtains. In 
this it will be readily assumed that those intermediate 
stages, for which the substituents stand in simple 
stoichiometrical ratio to one another, and for which 
a uniformity of distribution in the space-lattice is to 
be anticipated, will stand out somewhat from the 
others in, say, their optical or chemical properties. 
The researches of G. Tammann on isomorphous 
mixtures point to this conclusion. 

We may venture to conjecture that for these 
ratios the linkages in the force field are strengthened. 

Mixed crystals then have a specially important 
physico-chemical interest, owing to their explicit 
character as intermediate stages between chemical 
compounds and physical mixtures. 

The inner meaning of this is that chemical and 
physical methods of association are not so essentially 
different as has been assumed for some time in 
chemistry. 

Not uncommonly, a claim for the sharpest separa- 
tion of the two conceptions is made. Many, especi- 
ally old chemists and physicists, have, however, 
signified their acceptance of the same conception 
as is arrived at here. Of these may be named 
am bocrhave, C. L..Berthollet, J: B. Biot, J. C. Pog- 
gendorff, H. Kopp, Guldberg, and Waage, D. Men- 
deléef, H. J. van’t Hoff, and many others, and as 
W. Nernst, the master of modern chemistry, expresses 
it shortly in his text-book of theoretical chemistry, 
“the differences between physical mixture and 
chemical combination are really matters of degree, 
and we find in nature all gradations between the two.”’ 


102 CRYSTALS AND MATTER 


OUTGROWTHS WITH SUBSTANCES NOT ISOMORPHOUS 


This being so, these ideas may be extended 
even further to include the observed outgrowths 
according to definite laws which occur for various 
substances, e.g. rutile (TiO,) and hematite (Fe,O,), 
or potassium iodide and the complicated silicate, 
mica. When the particles are near in a leptonic 
sense, physico-chemi- 
cal reciprocal actions 
come into play which 
may lead to coupling 
up. It is convenient 
following F. Grandjean 
to speak of a “‘ champ 
moléculaire de contact.”’ 
As is known, crystallis- 
ing potassium iodide 
deposits on mica regu- 
larly oriented with 
respect to its mica sub- 
Ke Aa Tell ccvatals showitie 

FIG. 121.—Crystallisation of NaCland ‘ 
ar pee aa and separating tahedron surfaces, +4 
contrast to the cubic 
form of freely formed individual crystals. 

Apart from these macroscopic compounds, a 
series of phenomena may be obtained in macroscopic 
and microscopic experiments, which extend the idea 
of the formation of oriented crystallographic aggre- 
gates from the colloidal state down to the greatest 
dispersivity, i.e. to molecules. 

Thus chemical and physical modes of coupling of 
the particles converge in their essentials to the same 


FINE-STRUCTURAL PARTICLES 103 


thing. Strong and weak linkages are connected by 
intermediate cases, which can, even in the same 
substance, be followed through as gradations from 
the one extreme of chemical combination to the other 
of evident indifference. Certainly, in this generally 
important connection the frequent occurrence of a 
variation of the isomorphous miscibility with tem- 
perature is worthy of notice. NaCl and KCl, for 
example, at high temperatures, as they are obtained 
by crystallisation from the molten state, occur in the 
space lattice in arbitrary proportions, Na, K and 


i 1EN 
SZ; 
oy) 


Fic. Rare an of Fic. 123.—Parallel outgrowth 
rutile on iron glance. of KCl and NaCl. 


As crystallographic evidence of the weakening force field, and as symbolic 
forms of adsorption compounds (p. 130). 
Cl particles being then linked up in some regular 
fashion. On lowering the temperature this power of 
coupling up internally, possessed by the NaCl-KCl 
crystal fine-structure decreases steadily. The mix- 
ture separates out, and finally NaCl and KCI particles 
lie in contact with, but independently of, each other 
in a physical mixture.t The reciprocal action is not, 
however, entirely extinct at lower temperatures. 
On a freshly cleaved substratum of rock-salt a con- 
centrated KCl solution deposits small cubic sylvine 


1JIn Fig. 121 these relations are set out in a diagram for which 
we are indebted to the studies of Kurnakow, Shemtschushny, and 
Nacken. A. Smits gives also the complete system 
KC] — NaCl — H,O. 


104 CRYSTALS AND MATTER 


crystals in parallel orientation to the substratum, 
which is a definite indication that an actual chemical 
field is set up between the KCl and NaCl, under the 
influence of which the KCl particles arrange them- 
selves parallel to the NaCl space-lattice. In the re- 
verse case, also, NaCl will deposit on sylvine oriented 
parallel to the KCl space-lattice. 

In view of these examples, an important role 
must certainly be ascribed to temperature in deter- 
mining the capacity of aggregation in solution of the 
molecules. 


IX. MORPHOTROPY 


HISTORICAL 


HE particular type of fine-structural constel- 
lation in the molecule or crystal, depends 


naturally, in the first place, on the nature of 
the components. It is, therefore, of great interest to 
compare in this respect various substances with each 
other, and the most likely method of attack appears 
to be by chemical substitution. 

On these lines P. v. Groth has founded the science 
of morphotropy. By comparing crystalline forms, 
he succeeded in showing how the replacement of 
Hieby. (Ol); NO,, NH,, CH,, Cl, in benzene (C,H) 
manifests itself in changes of corresponding angles, 
and ultimately of the symmetry. It became possible 
to determine the morphological value of atoms and 
radicals, and by multiplying examples to lay down 
general rules in this respect. | 

Isomorphism, with its interchanges of allied sub- 
stances, say of potassium by rubidium in the sul- 
phate, presents itself here as isomorphotropy. Sub- 
sequently, as the result of work by W. Muthmann 
and F. Becke in particular, it was possible to inter- 
pret these morphological values with respect to 
definite physico-chemical masses such as the mol. 
The axial ratio a:b:c, with a different unit of 
length 6 = 1, for every crystal, was now replaced by 


the cm. values of the “ topic axes,’’ so that absolute 
105 


106 CRYSTALS AND MATTER 


comparison could be carried out.. Further, the axial 
lengths of the elementary cell can be employed as 
cell axes, and, finally, those of molecular and atomic 
regions as leptonic axes; this is actually done in a 
table which follows. The morphotropic variations of 
the fine-structure have been termed by A. Johnsen, 
topotropy. 

A brief statement such as the following gives a 
general idea of these important relations. 


STEREOCHEMICAL AXES 


Using the Loschmidt number N = 6:06 x I0*? as 
the estimated number of molecules in the mol (gram 
molecule) ! of any substance, it is possible to calculate 
the number of molecules in any other mass, and 
further, by a knowledge of the density) mass in 
grams per c.c.), the number in an arbitrary unit 
of space. Of these space units the elementary cell 
(a term due to P. v. Groth) interests us in fine- 
structure work; the reader is already acquainted 
with it in Fig. 24, page 22. Its dimensions are 
determined by X-ray methods. The number and 
also the weight of the molecules belonging to it can 
be easily calculated theoretically. 

The known cell volume, and the number u of 
molecules composing it, give immediately the portion 
of space appertaining to each of the m molecules of 
Preece 


1 By this is meant the number representing the molecular weight 
in grams of the substance. For NaCl the components Na and Cl of 
atomic weights 23:0 and 35:4é respectively, it amounts to 23 + 35-46 

* The weight of a mol Gy is connected with the volume of a 
mol Vm and the density S by the equation Gy = VmS. The mol 
axes are obtained in cm. by a simple calculation. In .crystallo- 


MORPHOTROPY 107 


Nothing now stands in the way of numerical 
comparisons of the fine-structure of crystallographic 
and chemically similar substances. 

As an example of such an investigation, the table 
for the closely related isometric salts of KCl, KBr, 
and KI is here given. Sodium chloride, NaCl, and 
the cyanide, KCN, are also included. In structure, 
all these are of the sodium chloride type (p. 22). 


Mol. Cell (cube.) Molecular Domain. a3 

sO 

Specific oe: 
sere At ase. | Volume | Weight | Axes | Volume | Weight | “%°S | Volume | Weight] Axes | 56 
1O-=: LO-— ame no 

C. Cm). 8 ann ord as ae eben Canin et Cal 53 

KCi 1-990 | 37°64 | 74°56| 3°346| 247-72] 492-96/ 6°280| 61°93 | 123°24 | 3-956] 4 
KBr 2°756 | 43°19 | 119:02| 3°508 | 285°52| 786:92| 6°585 | 71°38 | 196°73| 4-148] 4 
KI 3°134 | 52°97 |16 6-02 | 3°756 | 350-24 | 1097°64| 7°049 | 87°56 | 274°41 | 4°441| 4 
KCN 1-546 | 42°13 | 65:11 | 3-480 | 279-73] 432°30| 6°54 | 66°93 | 108-08] 4°117| 4 
NaCl | 2°173 | 26:90 | 58-46) 2-996| 177°58| 386°52|5°628| 44°47 | 96°63/3°543) 4 


In the series KCl, KBr, KI, with the same cation 
K+ and varying anion, the continuously increasing 
appropriation of space by the compounds, both as 
regards volume and axial lengths, is very definitely 
indicated, as is also the fine-structural equivalence 
of the group (CN) and the bromine ion. Sodium and 
potassium show their morphological action in a 
similar way. 


MORPHOTROPIC CONSTRUCTIONS 


¢ 


Making use of the results for “atomic domains ”’ 
(p. 46), fine-structures can often be predicted with 


graphy they are termed “‘ topic axes’”’ by F. Becke and W. Muthmann. 
Denoting the Loschmidt number by N, we have A ae Vir (Vi, = 


~ 


leptonic volume, i.e. volume of one molecule), and placing N molecules 
in the cell Z, Vz =7.V{. 


108 CRYSTALS AND MATTER 


surprisingly close approximation to the actual 
measurements. In a certain sense the spherical 
domains lie at hand as bricks to be built-up ina 
definite structural scheme. To take an example, the 
number d,, = 3°51 obtained as a mean from various 
calcium compounds, and the value d, = 1:26 from 
MgO lead to the cubic elementary cell constructed in 
the NaCl type, of CaO with an edge, a = 3:51 + 
1°20.= 4:77 X 10-8 cm.; X-ray value= 4°38 X 10m ae 

Further, let us suppose that we are dealing with — 
the still unknown fine-structure of rubidium chloride. 
The NaCl type is here to be expected with an elemen- 
tary cell of side 6-59 x I0~8 cm. corresponding to 
the assumptions dp, = 4:47 + do, = 2-12 X 107 
which will be approximately true. 

The simple relation holding for the atomic do- 
mains of allied substances makes it possible to in- 
vestigate, either graphically or mathematically, new 
types of atoms with regard to their atomic domains. 
In this way the series Me ++ = 2:99; Cat? ==3:5am 
ort + = 3:96; Batt = 4°36 leads to) thesragmams 
ion Ra++ = 4:90. Taking the domain diameter of 
OQ- ~ = 1:26, we arrive at the elementary Ccelma 
radium oxide Rat+t+O-~—, built in the NaCl 
pattern with presumably a cube edge of length 
rl, pec O we cas 

Even more complicated constructions may be 
carried out giving satisfactory agreement with X-ray 
and goniometric investigations. 

In this connection the trigonal crystalline salt 
CsCl . ICl may serve as anexample. In Fig. 124 the 
spherical atom domains of the elements named are 
piled up one on the other in close packing. Their 
dimensions (4d,, = 2°52 + dq = 2:12 + d, = 2°93 


MORPHOTROPY 109 


+ dq = 2°12 + $de, = 2°52) give the result 12-21 
hero, °* cm. X-ray research leads to the same 
number 12:2, certainly a good, if somewhat for- 
tuitous agreement, as regards accuracy. If with 
this close packing of spheres two material axial lines 
are constructed, the Cs ions touching those of Cl 
and I, the arrangement shown in Fig. 124 is obtained 
with an angle e = 39° 44’, which corresponds to an 
angle 41° 57’ in the goniometric measurements of 
the crystal. We might well rest content with this 
satisfactory agreement, but the fact is of crystallo- 
graphic interest that a very small elongation of the 
caesium domain in this trigonal, and therefore 
geometric rotational structure, gives the angle 
e = 42°, which the crystal actually exhibits according 
to goniometric observations. Fig. 125 illustrates 
this point. 

In Fig. 126 NaF .HF, as an example morpho- 
tropic to CsCl. ICl, is similarly constructed. Here 
also it is easy, by a slight alteration of the hydrogen, 
to build up a crystal form of the compound verified 
almost exactly by X-ray researches. 

Finally, reversing the argument, it appears pos- 
sible also to test a substance of known molecular 
volume with respect to its structural type. The 
case of the caesium salts CsCl, CsBr, CsI, may here 
serve asanexample. Their molecular volumes (mole- 
cular weight divided by specific weight) are, according 
to a table in P. Groth’s “ Chemischer Kristallo- 
graphie,” 41°80, 47:40, 56°85 respectively. If these 
molecular volumes are calculated for a body-centred 
elementary cell, the results come with convincing 
Seetoximation, tor) CsCl = 42756), (CsBri = 48-72, 
CsI = 58-70, indicating the existence of this type 


110 CRYSTALS AND MATTER 


of structure, while a construction in the NaCl scheme 
gives the widely deviating values 55:28, 63°28, 76-25. 


Fic. 124.— FIG. 125.— 
Construction of trigonal CsCl. ICI Construction of trigonal CsCl . ICI 
from spherical atom domains. using rotational ellipsoids for Cs. 


Fic. 126.—Construction of NaF .HF 
from atom domains. 


Moreover, recent X-ray experiments of Davey have 

shown that the former assumption is correct. 
Strictly, it must be supposed, according to 

K. Fajans and H. Grimm, that the space dominated 


MORPHOTROPY 1 


by an ion, say K+, will be affected by its particular 
Mech poursy apo itay Clasbrevor lode husiion no 
constituent in a compound is the atomic domain 
always exactly the same.! For the most part, how- 
ever, the relations are comparatively simple, inasmuch 
as it is possible, according to W. Biltz, F. A. Hen- 
glein, and E. Schiebold, to obtain graphically definite 
linear connections between the magnitudes concerned 
(p. 184). In short, we are on a very promising road 
leading to a knowledge of the activities of the par- 
ticles in the crystal microcosm, and consequently to 
the prediction of the properties of crystals. 


EXAMPLES OF FINE-STRUCTURAL MORPHOTROPY 
FROM THE MINERAL WORLD 

For the formulation of general laws every material 
is of the same value. At the same time, for those 
interested in natural science, there is a special attrac- 
tion in investigating the constitution and the laws of 
association of the substances which make up the 
structural material of the earth. Moreover, the fre- 
quently large size of such crystals is a favourable 
factor in the investigation; without the naturally 
occurring forms of quartz and calc-spar fundamental 
physical and chemical phenomena, such as double 
refraction, circular polarisation, etc., would certainly 
not have been discovered so soon. 

In the study of fine-structure, rock-salt, zinc- 
blende, lead-glance, calc-spar, magnetite, and several 

1 Hydrogen appears to be particularly sensitive in this respect. 
G. Aminoff estimated from the crystal structure of Mn(OH), and 
ice fyy = 1°15 X 10-5 cm. and 1:12 -++10—-% cm. respectively. 
Vegard obtained for NH,Cl and NH,Br, 0-83 and 0-84 x 1078 cm. 


respectively, whilst, again, NH,I and another modification of 
NH,Br gave rH = 1°12, or 0-99 x 108 cm. 


112 CRYSTALS AND MATTER 


more substances, have served for investigations 
which have become classical, whilst there remain 
in the collections of the mineralogist many likely 
materials for fine-structural investigation, such as 
quartz, the felspars, mica, augite, hornblende, gyp- 
sum and anhydrite, apatite, cinnabar, and numerous 
other mineralogical forms besides. 

With a view to adding to these examples from the 
mineral world, particularly in a morphological con- 
nection, I proposed to one of my students, Herr 
M. Mechling, that he should undertake, under the 
direction of my assistant, E. Schiebold, and myself, 


@) 
Fic. 127.—Topotropy for FeS, (iron pyrites) and CoAsS (cobaltite). 


a study of cobaltite, CoAsS, so that the results might 
be applied in a topotropical connection with reference 
to iron pyrites FeSs. 

The structure scheme for iron pyrites, according 
to investigations of W. H. Bragg and W. L. Bragg, 
is shown in Fig. 127: Fe at the corners and on the 
surface centres of the elementary cube containing 
four molecules, and of side 5:40 x 10-8 cm.; S, as 
a double atom placed centre symmetrically on the 
ternary axis. Considered topotropically, Co appears 
in the place of Fe, and an As appears for one S of the 
doublet in this equilibrium system, deforming it and 
giving rise to another stable arrangement. It is 


MORPHOTROPY 113 


found that the length of the cube side is only very 
slightly changed, namely, from 5-40 to 5-66 x Ico 78 
cm., the remaining S is displaced from its position on 
the body diagonal of the cube, and, at the same time, 
the substituted As does not take the position of the 
displaced S atom. The centre symmetry is lost 
on account of the leptonic inequality of As and $ 
in the new arrangement of the atoms. Cobaltite 
structure belongs not to the pinacoidal group, as does 
iron pyrites, but to the pedial group. All mirror 
symmetry in the fine-structure has vanished.! 

For the unusually numerous isomorphous mix- 
tures in the mineral world it is further possible not 
only to compare topotropically the extreme substances 
of the sequence of mixtures, but also to follow out 
the process of change, step by step, in the series. 
With this in view, Lotte Berndt, under the direction 
of Dr. Schiebold and myself, has worked on a par- 
ticular case. The mineral investigated was olivine. 
The elementary structural cell of this Mg,Si0, sub- 
stance (so-called forsterite) is a rectangular parallel- 
opiped. It is of considerable interest to see how 
this elementary cell changes when iron atoms are 
introduced for one part of the Mg in Mg,SiO,, by 
which process a so-called isomorphous mixture (Mg, 
Fe).SiO, (chrysolite) is obtained, and _ resulting, 
finally, in Fe,SiO,. The topotropic effect is brought 
out in the following table : ? 


1 According to later research on etch figures by H. Schneider- 
hdhn, there occurs for cobaltite a rhombic transformation modifica- 
tion with very tine twinned lamelle. The consequent mimetic 
isometric structure gives rise in the Laue diagram to an effect of 
correspondingly high symmetry. 

* In this az, tz, cz denote the edge lengths of the elementary cell. 

8 


114 CRYSTALS AND MATTER 


az bz cz in 1o—8 cm. 
Forsterite Mg,SiO, 4°74 10‘19 5:97 
Chrysolite (Mg, Fe),Si0, with about 
7 per cent. FeO . ; . 4°84 10°40 6:10 
Fayalite Fe,SiO, . ; : ~. 4°99 10°89 6°31 


It is indeed a triumph of physics that, thanks to 
X-rays, the absolute values of such small displace- 
ments in the microcosm can be measured. 


a WEARS OME SARA 


UnvolaliorolLeniny CRYSTAL CLASSES, CRYSTAL 
BORNMOCAS es LA DLULLY “ly PES 


HE fine-structure of atoms, molecules, and 
crystals may justly be compared with the 
astronomical megacosm. In both cases the 
question is one of statico-dynamical stability sys- 
tems. It is immediately obvious, then, that as 
certain structural arrangements recur in star systems, 
so also types. arise in minute structure. 

In the visible crystal world such isotypes are 
given in the crystal systems (syngonies) and crystal 
classes. 

The groupings in the table of page 31 must, in 
this way, be thought of as stability schemes, and it 
is probable that they could 
be calculated as a conse- 
quence of attracting and 
repelling forces. In this 
way new light is thrown 
on the crystallographic 
ideas stated at the be- 
ginning of this book, for 
there we had to deal with 
cases of isodynamostasy on Fic. 128.—Isotypic form 
a large scale. Further, the Be toate 
considerations may be extended to the specific gonio- 
metric ratios within the subdivisions. With this in 

115 


116 CRYSTALS AND MATTER 


view I have noticed the remarkable analogy in the 
crystal forms of very different substances, especially 
those of simple chemical constitution, such as the ele- 
ments, oxides, sulphides, halides, etc., in which certain 
architectural characteristics are repeated. For these 
the stability form of the complete isometric system 
is required; for trigonal substances the arsenic, 
quartz, and graphite type, as well as that of 
hexagonal magnesium. These crystallographic form 
groups stand in close morphological relation, as the 
plan in Fig. 128, page 115, indicates diagrammatic- 
ally. 


FINE-STRUCTURE GROUPS AS STABILITY TYPES 


Furthermore, by means of X-ray investigation of 
crystals, there is an opportunity of examining the case ' 
of isotypy with reference to the fine-structure. In 


1As examples of special isotypy the following hexagonal 

substances (magnesium type) are tabulated :— 
End surface : 
pyramidal 
I : 1°6391 62707 
I: 1°5802 > 61° 17. 
IT: 11-6554 627 235 
1 31'6288 62° 0” 
1 21-6216 61° 54’ 
I : 1°6305 62°25 
I : 1-6006 61° 35’ 
Greenockite, Cds , I : 1:6218 61° 54’ 
Magnetic pyrites, FeS Biles ove Biot Lay 62° 19’ 
Covellite, CuS . : ; S atiek Sono O12 At 

I 

I 

I 

I 

I 

I 

I 

I 

I 


a:c 


Magnesium, Mg 
Beryllium, Be . 
Cadmium, Cd : 
Irodosmine, (Ir, Os) . 
Zinc oxide, ZnO 
Beryllium oxide, BeO 
Wurtzite, ZnS . 


Arsenic-nickel, NiAs” : 16389 62°90" 
Antimony-nickel, NiSb Ee7 220 63° 18’ 
Silver iodide, on > 1°6392 62° 9’ 
Ice, H,O a oF hy 61° 50’ 
Tridymite, SiO, : : 16530 62°\2%0 
Cadmium iodide, Cals > 1°5940 OL 29) 
Lead iodide, Pbl, > 1°6758 62° 40’ 
Carborundum, CSi : > 1-6324 Nae a 
Copper glance (pseudo-hexagonal), Cu,S : 16707 02” 30. 
Chrysoberyl (pseudo-hexagonal) 

BeOA1LO;  . . : : <) Pt Oat 61° 55’ 


ISOTYPY 117 


fact, the correctness of the idea becomes more and 
more obvious, for certain structural schemes occur in 
remarkable abundance, and in peculiarly close archi- 
tectural relation to one another. For the examples 
mentioned up to now, the case of a surface-centred 
elementary cell with its tetrahedral placing of the 
atoms in the form of the isometric type for elements 
and simple compounds, occurs very frequently in- 
deed. This corresponds to a very stable style of 
structure, which still stands out prominently on 
trigonal deformation. On closer consideration of 
the elementary cells (Fig. 129 and 24, p. 22), the 


Fic. 129.—Lattice type of rock-salt and its trigonal deformation to the calc-spar 
type. 


pre-eminent importance of the tetrahedral grouping 
will be easily recognised as parts of the sections there 
depicted, not only for diamond and zinc-blende, for 
example, but also for copper, rock-salt, fluor-spar, 
and in the deformed calc-spar. In Fig. 130 tetra- 
hedral types of structure are reproduced to show 
their special character. 

In view of the isotypic agreement in the stability 
of the isometric diamond and zinc-blende fine- 
structure, it was of great interest to see if this equili- 
brium form transmits itself to any extent to the 
hexagonal type, a form I have ascribed to mag- 
nesium, and to which carborundum consisting of C 


118 CRYSTALS AND MATTER 


and Si also belongs. 


The attractive problem of 
investigating the fine-structure of this carbide by 


means of the delicate and yet so powerful agency of 


3, 
Fic. 130.—Examples of tetrahedral structure: Copper, rock-salt, diamond, 
zinc-blende, iron pyrites, calc-spar. 


. 
6 
ome 
xm 


+ y,". ¥ 


rs 


ae.’ 


Fic, 131. Fie. 132. 
Fics. 131 and 132.—Family relationship between the Laue diagrams of a 


diamond twinning (plate parallel to the octahedral surface) and of carborun- 
dum (plate parallel to the end surface). 


X-rays was undertaken and carried out with great 
success by H. Espig, under the direction of Dr. E. 
Schiebold and myself, on type II. of carborundum. 


ISOTYPY 119 


I had already earlier pointed out, in a comparison 
of the Laue diagrams of diamond and silicon carbide, 
the great resemblance of these spectral symbols of 
the fine-structure ; this is brought out well enough 
in Figs. 131 and 132. As the crosses in the figure 
for carborundum show, it contains all the reflexions 


Fic 133.—Fine-structure of modification II. of carborundum. C, dark; Si, 
light circles. 


of the diamond. The detailed X-ray study of 
H. Espig very clearly indicated the correctness of 
the assumption of a structural affinity. The tetra- 
hedral diamond structure is involved to some extent 
as a component of carborundum. Its carbon atoms 
constitute goniometrically a form nearly identical 


120 CRYSTALS AND MATTER 


with those of diamond. In this way, one of the two 
(for the diamond, equal valued) carbon families is 
reproduced here, the Si atoms of the carborundum 
replacing the second tetrahedral group of the dia- 
mond. Here, however, an important change in 
configuration occurs, in that this type of atom is 
arranged not in tetrahedra, but in slender trigonal 
pyramids, which are set with the apex of one in the 
body of another. Moreover, the rearrangement of 
the C and Si stars in this complicated heaven is also 
shown in the variation of the side length of the car- 
bon tetrahedron ; it amounts in 
the diamond to 2°5 x 1078, and 
- In carborundum to 3°I x Io~7 8 
cm. Thus both the diamond and 
Fic. 134.—Tetrahedral fine. Carborundum belong, crystallo- 


structure of hexagonal zinc 


oxide. Large atom domain graphically, to a special type, 
(d= 2:64)Zn, small, (¢=1'26)0. and it is now definitely recog- 
nised that they are in consequence intimately con- 
nected fine-structurally. 

The same thing applies to hexagonal zinc oxide, 
which W. L. Bragg has investigated. According to 
him its oxygen atoms form everywhere the middle 
points of tetrahedra whose corners are occupied by 
zinc atoms. Hexagonal zinc oxide stands, therefore, 
in close relation to isometric zinc sulphide, and also 
to the diamond.! According to G. Aminoff, an 
analogous case in connection with isotypy arises for 
Mg (OH), and H,O. In its formal fine-structure, 
Mg(OH), can be regarded as H,O.OH, structure, 
for which one H, has been replaced by a magnesium 
atom and spacial condensation has occurred. 


1Isometric ZnS shows in its Zn and S tetrahedral groups the 
two tetahedra of the diamond (compare Fig. 24d and 24e, p. 22). 


XI. CRYSTAL GROWTH AND SOLUTION 


HE conception of the crystal as the stability 
form of attractive and repulsive anisotropic 


forces, involves the assertion of its reaction 
to external physical or chemical changes in order to 
accommodate itself to the new conditions. 

In this sense we may consider the extensibility 
and compressibility of crystalline materials, and 
many other physical properties, under the influence 
of temperature or pressure change. The capacity 
of crystals to react to their surroundings stands out 
as specially obvious in the easily observable pheno- 
mena of growth and solution. These may therefore 
be briefly considered here. 


PURE CRYSTAL GROWTH 


Every crystal has grown from a tiny nucleus, and 
in this process of enlargement by addition of succes- 
sive shells of parallel placed particles, the anisotropic 
character of the substance is very clearly shown. 
In this way we get, in general, not a sphere, but a 
faceted body, indicating that the nucleus grows with 
different velocities along different directions, in the 
form of a growth pyramid; the crystal is thereby 
divided up genetically, as F. Becke first emphasised. 
Directions of similar growth recur at intervals. 

According to the fundamental work of A. Johnsen, 


which was extended by R. Gross and others, the curve 
I2I 


122 CRYSTALS AND MATTER 


of growth velocity, which is obtained by drawing pro- 
portional vectors from some fixed point in accordance 
with this growth anisotropy of 
the crystal, shows maxima and 
minima, and, moreover, very 
sharp gradients. Null or 


_ Fic. 135.—Growth by deposi- Fic. 136.—Growth pyramids. 
tion of successive shells (example 
of quartz). 


{219 


Fic. 137.—Intensity curve for the growth of rock-salt. 


infinite extremes do not occur. Normal to the minima 
directions, the large crystal surfaces naturally unfold 


CRYSTAL GROWTH AND SOLUTION 123 


themselves, since these remain near 
the origin. To them is to be as- 
cribed a lower surface energy com- 
pared with the faster-growing sur- 
faces. 

F. Haber, F. Paneth, P. Niggli, 
and others have represented the 
outer zone of the crystal as un- 
saturated with respect to valency. 
It attempts, therefore, to reach equil- 
ibrium by addition of new particles. 
Harmony with the outside is not 
attained, for a new surface is there, 
and the growth still continues. ey hie 

P. Niggli made a considerable of growth. 


©. @, O' @: 0 @€ 0 .¢ CO. €@:. 0 6. 0 


e©oe 0 © 0 & 0 @ 0 @ 0 Ox 
(ei0 


o-Altoms B=CL 
e=/iltoms A-Na 


On @ ON. 07 0) 10-20 OY On 6 
eye 0 ® 0 © 0 @ 0 @ 


De O— O: (Oia Os. O-- 0-20 40:0 
DN 


<3 0 60" - 0: 80; 50-0 ...6- 0 


Fic. 139.—Surface zone of unsaturated valency for rock-salt according to 
P. Niggli. 


124 CRYSTALS AND MATTER 


advance in this matter when he assumed the growth 
velocity proportional to the thickness A of the as- 
sumed unsaturated crystal layer, a magnitude differ- 
ent for different surfaces. In this way he arrived 
at theoretical growth intensity curves, which, with 
lead-glance and rock-salt (Fig. 139), for example, 
agree very well with our ideas derived from the 
morphology of these minerals. 

Valeton has developed some very illuminating 
views on the growth of crystals, with special reference 
to the structure of rock-salt from sodium and chlorine 
ions. The deposition of new particles depends, he 


FIG, 140. Fic. 141. Fic. 142. 


Fics. 140-142.—Ion groupings on the cube and octahedron surfaces of rock- 
salt. After Valeton. 


says, on the fine-structural nature of the surface. 
The chess-board aggregation in the cube planes of 
positive Na and negative Cl atoms is very clearly 
differentiated from the uniform octahedral surfaces 
containing either sodium or chlorine (Figs. 140-142. 
See. also, Hig. 27, p. 26): . The vformer ares 
favourable to the retention of an impinging ion of 
the solution than the latter. An ion which is to be 
retained on a cube surface must strike almost exactly 
in the middle of a field, whilst for the octahedral 
surface it depends on the sign of the ion, there being, 
on the whole, a 50 per cent. chance. The cube will 
therefore grow more slowly than the octahedron, and 


CRYSTAL GROWTH AND SOLUTION 125 


will, in consequence, predominate in rock-salt. An 
important point to be noticed in this connection is 
that whilst with the Niggli idea cube the octahedron 
and rhombic dodecahedron surfaces do not differ in 
the thickness of their unsaturated zones (zero in each 


aV2= 7292 
2 Qa =9 ] 23 9 #4, 


Fic. 145a. Fic. 1450, 


Fics. 143-145.—Surface structure of the cube, rhombic dodecahedron and 
octahedron surfaces of rock-salt, indicating the arrangement of the atom 
domains. (The second scale relates to depth distances.) 


case), on Valeton’s views there arises the possibility of 
growth differences for these surfaces. In this respect 
the great dissimilarity of the surfaces may be indi- 
cated in greater detail by Figs. 143-145. These show 
fine-structural aspects of such surfaces, indicating not 


126 CRYSTALS AND MATTER 


merely the external plane but also the region beneath 
by representation of the atomic domains. The action 
of the attractive forces, which presumably are capable 
of making their influence felt over several periods of 
the fine-structure from the inside, outwards, will for 
a cube surface be screened off by packs of only two 
layers by the covering of the atomic domains.! 
Matters are otherwise for the octahedron, or even 
for the pyramidal cube surfaces, as Figs. 144 and 
145 demonstrate. 

Certainly these fine-structural differences with 
respect to the region below the surface, together 
with the net density (as the measure of the number 
of particles belonging to the layer), are of immediate 
significance as obvious explanations of the variation 
of growth with direction in the crystal. Further, it 
must be remembered in all these considerations that 
in crystal growth we are concerned with the action 
of a complicated chemical field between the crystal 
and its surroundings, otherwise there would be no 
possibility of explaining the great influence on the 
crystallisation of the other things present in the 
solution, which can give to sodium chloride, in one 
case, the form of a cube; and in another case, that 
of an octahedron or a pyramidal cube (Fig. 146).? 


1 Here, as in other cases as for diamond (p. 55), and still toa 
considerable extent in zinc-blende, canals as penetrating rifts in 
the structure persist in these constructions which traverse the 
crystal in certain directions, forming a check pattern. Their signifi- 
cance, taken together, as a porosity of the crystal, remains to be 
investigated in another place. 

? We might assume for this that the surface of the crystal is 
chemically compensated by a special arrangement of the particles 
there, and under the influence of the solution immediately rearranges 
itself (compare p. 79). 


CRYSTAL GROWTH AND SOLUTION 127 


In consequence, such additions to the solution 
must possess the power of altering the factors con- 
ditioning growth velocity at the crystal surfaces. 
For example, the growth normal to the octahedron 
surface may be retarded, and this external form 
imparted to the crystal. Such, more or less, effective 
protection of the surface will be explained by the 
adsorption of the other substance present in solution, 
an explanation proposed by R. Marc. Experiment 
very definitely supports his hypothesis of a retarda- 
tion of growth by adsorbed substances. With 
calcium sulphate, for example, on addition of merely 


a b c 


Fic. 146.—Rock-salt crystal: a, from pure solution; 8, from solution containing 
glycocoll; c, from solution containing formamide. 


a trace of a certain dye, which is absorbed by the 
crystal, we get, in contrast to the well-formed crys- 
tals obtained from pure solution, an irregular mass 
of thin sheets. When the crystal surface is satu- 
rated adsorptively it loses the power of acting as 
a nucleus. Such crystals may be shaken up in 
supersaturated solution for days without disturbing 
the supersaturation. 

The particular form of a crystal constitutes a 
morphological symbol of the equilibrium of the 
balanced force fields which arise between its own 
substance and the materials around. 


128 CRYSTALS AND MATTER 


MIXED CRYSTAL GROWTH 


For the analytical chemist crystallisation and 
crystal growth have a special importance. Crystal- 
lisation is for him a process of molecular selection, 
inasmuch as one kind of matter separates cleanly in 
crystal nuclei, on which more of the same substance 
is then deposited. This process brings to mind ina 
general way the ability of organisms in an abundance 
of material, say, of a solution, to apply it to the 
crowth of a certain one. 

Of considerable general interest in physical 
chemistry, and of significance to the analytical 
chemist, is a circumstance which may diminish the 
value of crystallisation as a means of purification. 
If, in the chemical field between crystal and solution, 
besides the substance chemically identical with the 
crystal, there exist also others, which, although 
different, are chemically analogous to it, as, for 
example, besides the chloride of potassium, those of 
Rb or (NH,), then interchange of atoms or radicals 
may occur. 

This has been known as an experimental fact 
since the researches of J. N. Fuchs and F. E. Mit- 
scherlich, and later, in the light of space-lattice 
theory, has become very clearly understood, fine- 
structurally.1. The case here is one of the growth of 
isomorphous mixtures (p. 97). Moreover, different 
mixtures, or the pure extremes of substances so 
related, are able to build themselves up into layers. 
A splendid example of this is shown in Fig. 148 for 
zonal tourmaline. | 

The inclusion of such alternative structural units 
in the growth can naturally only take place in so far 


1 See Fig. 147. 


CRYSTAL GROWTH AND SOLUTION 129 


as the stability of the whole is not imperilled. Often 
that is not at all the case for extensive isomor- 
phous mixtures as for felspar, olivine, and other 
minerals. In other cases, warpings of the form or 
optical anomalies point to discordance in the structure 
of such heterogeneous lattices. There exists also a 
certain danger with respect to the architectural 
cohesion, that on change of the temperature the 
alternating particles may lose more and more their 
structural equivalence by the swinging round of 
their electrons, i.e. by change of form. A pair of 


FIG. 147 .—Isomorphous alternation Fic. 148.—Isomorphous stratification 
of Cl and Br in the space-lattice. Example: tourmaline. 


substances such as Na and K is simply incapable of 
this alternating incorporation in the growth, unless 
the atomic structures are first in a similar condition. 
This leptonic isomorphism depends on the physico- 
chemical factors, temperature, and pressure, as well 
as on the action of the surrounding material, these 
being the effective influences on the fine-structure. 
In this respect the marked change in the miscibility 
with temperature of NaCl and KCl already men- 
tioned on page 103, is of interest. 

Especially worthy of note in these phenomena 
of regular incorporation in mixed growth are those 
2 


130 CRYSTALS AND MATTER 


cases in which occasionally a substance quite foreign 
is incorporated. Many minerals bound together 
macroscopically in regular fashion, such as cyanite, 
staurolite, rutile, and iron-glance exhibit the relation- 
ship (Fig. 122, p. 103). For very small dimensions 
of the definitely oriented guest material, and for more 
extensive growth of the surrounding host, a kind of 
colloidal solid solution with regular packing of the 
substances, but without any stoichiometrical ratio 
one to another, is arrived at as, 
for example, in “ adsorption com- 
pounds,” “occlusion,” and the 
like. 

As to the forces of cohesion 
between the crystal particles and 
such foreign bodies, no deviation 
will be made from the hypothesis 
of electrical coupling. This hy- 
Ser ieee pothesis has continually increas- 
oe paint eee) ing support, as, for instance, in 

connection with the flocculation 
of colloidal substances. There again it is possible to 
employ crystals as types with actual visual observa- 
tion. Electrically bound charges may, in fact, be 
immediately observed for pyro-electric crystals like 
quartz, which on change of temperature becomes 
electrified positively and negatively respectively on 
its alternate column edges (Fig. 149)... To show the 
distribution of the oppositely charged regions in such 
cases, A. Kundt employs a mixture of sulphur and 
red lead which has become electrified by being blown 
out of a suitable sprinkling apparatus. The positive 
red lead sticks to the negative parts of the crystal, 
coating them red, the negative sulphur deposits on 


CRYSTAL GROWTH AND SOLUTION 131 


the positive regions, which consequently appear 
yellow. In this way an adsorption coupling of 
crystallographic nature can be immediately detected 
for quartz. 


COLLECTIVE CRYSTALLISATION 


It is further of great physico-chemical interest 
that forces tending to form aggregates exist not only 
between crystal and solution, but also between 
crystal and crystal. In fact, it is certainly one of the 
most remarkable phenomena of the crystal world 
that in spite of the rigidity of the material there is 
this tendency to collect together to larger individuals, 
1.e. the space-lattices of neighbouring crystals tend to 
set themselves parallel and to link up; this process, 
which I have studied in various cases, I have termed 
“collective crystallisation,’ which indicates _ its 
nature. The effect is, in view of the coalescence of 
small particles to form large ones, a “ coarsening of 
the grain.’”’ The process may be demonstrated in a 
few minutes by heating up cast steel. Every chemist 
is acquainted with it in the much-used platinum 
crucible, and the technical ighting workman, in fine 
tungsten filaments, the innumerable small crystal 
individuals of which, on glowing, are turned round 
into single crystals which may be a metre in length. 
Tungsten trioxide also is convenient for showing the 
phenomena, and calc-spar too, if care is taken that in 
the intense heating the CO, is not allowed to escape, 
as is the case if the experiment is carried out in 
a carbonic acid bomb. The marmorisation of dense 
limestone in contact and regional metamorphosis is 
also a case of this collective crystallisation.! 


1 What happened here was a transformation of ordinary dense 
limestone into marble by molten masses, perhaps of granite type, 


132 CRYSTALS AND MATTER 


In such cases of closely packed, new-grown struc- 
tures (as Figs. 150) and 151) show), the individuals 


Fic. 150.—Collective crystallisation for iron: a@, Martin steel; 6, the same 
Martin steel glowed in the furnace. 


Dare 
iors, s 


ROM 
on; 


‘ z “ 
XK MG SI 
At fi ; 
Late tb) sed! S (ado 
WS Coe IS Kea 
Soars Keg 
oa iret 
dew AAS 


Fic. 151.—Collective crystallisation of calc-spar: a, limestone; 4, marble. 


hinder each other in the production of regular 
crystallographic forms. If, on the other hand, they 


which, arising from the depths of the earth, stuck fast in the stony 
crust, and then, by reactive action of the gases and liquids liber- 
ated, and the high temperature, extensively transformed the matter 
around. Here, at the “ regions of contact,’’ collective crystallisation 
played a great part, and, in addition, many substances finely dis- 
seminated in the original material became associated into large 
crystals of, perhaps, graphite, andalusite, augite, garnet, etc., so 
that here again collective crystallisation was effective. 


CRYSTAL GROWTH AND SOLUTION 133 


swim around freely in a liquid, association by collec- 
tive crystallisation, with new development of crystal 
surfaces, can proceed. This is the case for the 
highly interesting combination of contiguous ammo- 
nium oleate crystals described by O. Lehmann. 
Elongated pyramidal individuals coalesce to a struc- 
tural unit more or less definitely crystallographic in 
contour, or, at least, to a group of crystals with their 


Fic. 152.—Collective crystallisation for ammonium oleate. After 
O. Lehmann. 
principal axes parallel (Fig. 152). Moreover, it may 
probably be assumed that in every crystallisation, to 
begin with, numerous sub-microscopic crystals arise 
which form by aggregation visible crystal nuclei in 
the above manner. 


CRYSTAL SOLUTION 


Corresponding to many of the observations on 
crystal growth, a knowledge of crystal solution has 
lately been developed to a very gratifying extent. 
The observations here indicate that the principles of 
the phenomena are closely connected. It appears of 
use in dealing with the solution of crystals, as in the 
discussions on their growth, to employ the very clear 
picturisation of the atomic domain stereograms. In 


134. CRYSTALS AND MATTER 


this way we see immediately, as the examples of 
Figs. 143-5, page 125, show, the anisotropy which 
occurs both in solution and growth. It becomes clear 
that the displacement of the surface, by the attack 
of the solvent on the crystal structure, takes place 
with varying rapidity, depending on the direction of 
attack. Research definitely corroborates this state- 
ment, to which the pretty phenomena of etch figures 
bear immediate witness. These are formed in large 
numbers, and mostly of microscopic dimensions, on 
crystal surfaces attacked by a solvent, and represent, 
in the form of cavities or eminences with regular 


Fic. 153, a, 6, c.—Etch figures and the corresponding light figures on a side 
surface of gypsum, together with figures for the same mineral dehydrated. 


edge and surface boundaries the symmetry of the 
crystal faces on which they occur (Fig. 153a). On 
examining all the results so obtained, the structural 
style of the whole crystal body becomes evident. 
With the reflexion goniometer corresponding light 
figures may be measured (Fig. 153b). In both figures 
the binary character of the surface is brought out. 
Good examples of solution anisotropy are very neatly 
shown by plates cut into circles, an example of which 
is shown in Fig. 154. It represents an initially 
circular plate of gypsum, which on immersion in 
water has changed to a pointed figure of elliptic 
periphery, owing to the anisotropy of the solution 
velocity. The relations of polished crystal bodies of 


CRYSTAL GROWTH AND SOLUTION 135 


regular, and in particular, cubic form, or of crystal 
spheres, are known in even greater detail. Growth 
and solution velocity mutually correspond, the repre- 
sentative vectors rising and falling together. The 
directions of rapid growth are those of rapid solution, 
and vice versa. During the solution of a crystal a 
struggle takes place between the anisotropic velocities 
of solution. The surfaces, which quickly approach 
to the mid-point of the crystal, and which are those 
having large solution velocities in the direction of 


© > 
Uy Lo y oh i y / 
100 ox UY Ny, 
nM 
Yi il S 
403 

Fic. 154.—Anisotropy of crystal Fic. 155.—Reproduction of a per- 
solution. Example: Initially circular manently conformal solution form. 
plate of gypsum in dissolution. Example: Anhydrite in sulphuric acid. 


their normals, will gain more and more area, sup- 
pressing the slower ones which will probably be 
present at the start, and will finally form the 
boundary planes of the residual body. The scheme 
of Fig. 155 depicts such a final form for anhydrite. 
In the same connexion several figures from a research 
of W. Schnorr on rock-salt may be mentioned. 
Specimens of the mineral which are initially cubic 
(Fig. 156) assume the forms shown in Figs. 157, 158, 
159, 160, when immersed in an unsaturated solution 
of sodium chloride containing urea. At first there 


136 CRYSTALS AND MATTER 


are formed, on the cube edges, surfaces of the pyra- 
midal cube, which suppress the original form, but 
are then suppressed by those of the icositetrahedron. 
The icositetrahedron so produced is quite unaltered 
in shape on further solution. A stability form as the 
expression of the equilibrium attained is arrived at 
in this way. Under constant conditions, during solu- 
tion, as also during growth, the velocity of displace- 
ment of a surface is found to be always the same. 


Fic. 158. Fic. 159. . Fig. 160. 


FIGs. 156-160.—Solution process for a rock-salt cube in an unsaturated solution 
of sodium chloride containing urea. After W. Schnorr. 


An analogy between growth and solution also 
arises in this way, since, in both cases, the question 
is one of definite reciprocal action between crystal 
and its environment, depending on the nature of 
the other substances present, 1.e. on the chemical 
field. The morphological action of an anisotropic 
solution depends not only on the crystal but also 
on the particular solvent. This may be very clearly 
shown for anhydrite, which W. Burckhardt has in- 
vestigated on this point at my suggestion. Accord- 
ing to whether sulphuric acid, nitric acid, water, 


CRYSTAL GROWTH AND SOLUTION 187 


or salt solution was employed, different solution forms 
were obtained from a cubic cleave of the rhombic 
mineral (Fig. 161). 

The symmetry of the solution body is naturally 
the same in all states of its formation and for every 
solvent. It is identical with that of the growth 
figure. From the phenomena of solution the par- 
ticular class of the thirty-two crystal classes to which 
a substance belongs may be determined. 


ie oO 


(100} i{010} {001} {100}; a {100} ;{010};{ o01}i{0.1.10} {100}; {010};{007} ;{so%} 
is d 


Fic. 161.—Variation of the solution form for different solvents. Example : 
Anhydrite. a, Tri-pinacoidal initial cleave ; 4, solution in sulphuric acid ; 
¢, in nitric acid or water ; d, in salt solution. 


SUMMARY OF CRYSTAL GROWTH AND SOLUTION 


In a general survey of the processes of crystal 
growth and solution it must be considered that the 
question is, broadly, one of the displacement of the 
boundary between solid crystalline and liquid (in 
certain cases, gaSeous) masses which are exerting 
forces on each other in opposite directions. In 
growth the attractive forces of the crystal prepon- 
derate ; in solution those of the surrounding matter 
are greatest. The external surface is correspondingly 
displaced. All circumstances which diminish the 
electrical connection of the crystal particles act in 
the direction of solution, and vice versa. Water, 
with its high dielectric constant, and consequent 
extensive diminution of the electric attraction, brings 
many crystals to destruction. 


138 CRYSTALS AND MATTER 


Such crystals dissolve as soon as the conditions 
of temperature, i.e. of the motion of the particles, 
and the modifying influence of other substances in 
the solution, permit a displacement of the surface 
inwards as the result of the opposing forces acting in 
this direction. For insoluble substances the internal 
cohesion preponderates over physico-chemical forces, 
tending to break up the structure. 


XII, CHEMICAL ACTIONS ON CRYSTALS 


ANISOTROPY OF CHEMICAL ACTIONS ON CRYSTALS 


and chemical action are dealt with separately, 

no fundamental contrast in the processes is 
implied. On the contrary, they are very closely 
related. Thus the reaction CaCO, + 2HCl = CaCl, 
+ H,O + CO, for calc-spar manifests its regular 
anisotropy not only in pretty microscopic etch 
figures (and also light figures) 
(Fig. 162), but also in the varying nea So 
amounts of CO, developed in unit 
time on the morphologically dif- 
ferent surfaces. The same thing 
applies for the reaction body. Fig. 
163 gives a beautiful example of ae 
this. Fig. 164 depicts such a jy 162.—Rtch figures on 
case indicating the variation with  “le-spar crystal. 
change of the corrosive agent employed ; it refers to 
the decomposition of a tourmaline by caustic potash 
and hydrofluoric acid, which was recently studied in 
my institute by Lotte Kulazewski. The reactions 
with the silico-borate in question are especially inter- 
esting, as, owing to the trigyric domatic character of 
the mineral, they proceed vectorially (differently in 
the upper and lower halves of the tourmaline sphere), 
as a comparison of the diagrams Figs. 164a and 6 and 
c and d indicates. Figs. 164a and c represent the 

139 


A LTHOUGH the phenomena of growth, solution 


a 
aN 
m4 
ie | 


~ferrr--ccn 


140 CRYSTALS AND MATTER. 


upper, Figs. 6 and d the lower hemispheres with 
respect to the light reflexion of the reaction body ; 
both hemispheres are depicted up to beyond the 
equator. On allowing the reaction to proceed 
further, the vectorial character of the chemical 
process is realised morphologically with extraor- 
dinary effect, in that the sphere becomes more and 
more flattened on one side, and finally is transformed 


into a dome-shaped body, which has already been 


illustrated in Fig. 79¢, page 56. In cases of centre 
symmetrical structure the chemical action is ten- 
sorial, i.e. equal in any one direction, and the 
corresponding opposite direction. 


ANISOTROPIC CHEMICAL REACTIONS OF MOLECULES 


For individual molecules in gases or liquids the 
morphological principle of chemical reaction cannot 
differ from that for crystals. Indeed, the funda- 
mental conception of the stereochemistry of chemical 
reactions, such as the localisation of the replace- 
ment of one or more hydrogen atoms in the benzene 
molecule by Cl, NH,, CH;, or other radicals,! corre- 
sponds to this assertion. The process of chemical 
action, according to this, is always definitely aniso- 
tropic, and the action for molecules only differs from 
that for crystals in the suppression of a restricting 
rhythm in the structure, and consequently in the 
chemical relations. A centre or mirror symmetrical 
arrangement, or a “ plurality’ of places liable to 
attack may naturally, however, play a réle in chemical 
operations with molecules, inasmuch as completely 
similar topical relations (such as the two (NH,) groups 


1 For the mechanics of such processes, compare page 158. 


Fic. 163.—Reaction body of a sphere of topaz after treatment with caustic 
petash. After MW. Eichler 


Fic. 164.—Reaction anisotropy with ternary rhythm obtained by treating a 
tourmaline sphere with 1 (left-hand figure), caustic potash ; a, top surface; 
6, under surface. 2 (right-hand figure). hydrofluoric acid ; ¢, top surface ; 
d, under surface of the sphere. (Vectorial anisotropy of chemical reaction.) 


After Ch. Kulaszewskt 


CHEMICAL ACTIONS ON CRYSTALS 141 


in (NH,), CO; molecule) may show in similar reactions 
their equal fine-structural and equal chemical rank in 
the complex. 


STRUCTURAL RIGIDITY OF ELECTRONS, ATOMS, 
MOLECULES AND CRYSTALS 


According to the above, an estimate of the 
chemical nature of substances and of their changes 
turns on the investigation of fine-structure and its 
variations. This can only be carried out completely 
by reference to the stages in the graduated series 
which extends from the simplest form of matter, 
from electrons to atoms and molecules, and finally 
to crystals. 

With respect to the possibility of initiation of 
chemical reaction and its continuance, the general 
structural rigidity of the particles is the question 
of prime importance, inasmuch as the resistance 
to chemical change is dependent on this factor. As 
the particular circumstances vary considerably, we 
are interested here in the general state of affairs, 
that is, whether, for the purpose of the present con- 
sideration, the graduated series from electron to 
crystal is arranged in accordance with the structural 
rigidity. An examination of the series from the 
lowest to the highest member indicates unmistak- 
ably an increasing fine-structural complication, and 
parallel with this, a decrease in the rigidity of the 
various types. 

The architecture of the electron has remained up 
to now imperturbable. Such forms are, we may 
say, fortresses still untaken. 

Practically the same thing holds for the atom as 
for the electron, as far as the central portion, the 


142 CRYSTALS AND MATTER 


nucleus is concerned. A variation of this inner 
structure, the main mass of the atomic system, is 
only possible by enormous expenditure of energy. 
As we know, Rutherford has managed to cause disin- 


tegration of nitrogen atoms N into 2H+ and 3Hett 
by powerful bombardment with Het* particles. 
Only one out of every 100,000 shots resulted in 
an actual collision. Although, therefore, the nucleus 


of the atom is not indivisible, it is a structure ex- — 


tremely difficult to split up. 

For the outer shell the case is otherwise. As 
regards this external zone, the atomic system does 
not, in point of fact, accord with its name. On the 
contrary, the atom shell is quite easy to split up, 
and is thus ‘‘eutomic.’’ Since the acceptance of 
Sv. v. Arrhenius’ conception of ions, and following 
that, the fine-structural explanation of their forma- 
tion, as a splitting off of electrons from the outer 
sphere or an insertion of the same, the matter has 
become one of easily effected superficial variations. 
The idea of structural variations in the atom is made 
use of when light emission and absorption are ex- 
plained as the transposition of an electron from one 
stable path to another, with consequent loss or gain 
of energy quanta. In such cases quite small quanti- 
ties of energy render possible very real changes in the 
outer sphere of the atom. 

It would be superfluous here to go into details 
concerning the disruption and aggregation of mole- 
cules and their transformation by substitution ; this 
is, in fact, the main topic of chemistry. The incon- 
ceivable abundance of these phenomena shows how 
such changes can take place, with varying energy 


CHEMICAL ACTIONS ON CRYSTALS 143 


exchanges, which, in general, occur with relatively 
greater facility than in a physical field (e.g. by a 
rise of temperature or through the action of matter). 

Passing through the structural series to the final 
and most complicated forms, namely, crystals, which 
are specially considered here, the relations point in 
the same direction of readier variation in the archi- 
tecture. With crystals it is quite usual for the struc- 
ture to collapse on application of physical or chemical 
influences, and for the matter to assume a lower 
structural type. For instance, in the action of HCl 
on calc-spar, the space-lattice complex of the latter 
is broken up with the formation of free molecules, 
CaleH.,O and CO,. 


CRYSTALLOGRAPHIC CHEMICAL CHANGES IN THE 
STRUCTURE. UNDERMINING AND RECONSTRUCTION 


In less frequent cases, however, chemical changes 
occur in crystalline materials without destruction of 
the crystal structure, i.e. with preservation of the 
high crystalline status of the substance. We are 
then concerned with a particularly important case 
of the so-called topochemical reactions of V. Kohl- 
schiitter, that is, with a structural undermining. 
Further, it is occasionally possible to reverse the 
process or to substitute something else for the sub- 
stance removed, and thus to transform the crystal 
structure without at the same time destroying the 
crystalline character of the material. But naturally, 
these are extreme cases which are linked up by inter- 
mediate examples of more or less drastic disturbance 
with the other extreme of complete destruction of 
the crystalline form during chemical reaction. 

It is profitable to compare a process of ideal 


144 CRYSTALS AND - MATTER 


crystallographic chemical undermining with the par- 
tial destruction of a framework structure from which 
the filling between the beams has been removed. 
On account of this, the structure becomes less com- 
pact without, however, collapsing, and the principal 
structural lines are stid maintained. The simile may 
be readily extended to a reconstruction. The beams, 
although remaining in position, may, in the partial 
disruption, be damaged to some extent by splitting. 
and transverse fracture, and as structural particles 
in an undermining or reconstruction may become dis- 
placed, the stability of the whole may be impaired and 
finally lost. 

In the light of space-lattice theory, it may be 
assumed for the ideal case of undermining that from 
a point system such as the one of Fig. 9, page Io, 
one space-lattice is removed without the remainder 
collapsing, although it may be deformed to a new 
equilibrium arrangement. In reconstruction compen- 
sation is made for the removed lattice. We must, 
however, expect many gradations of the above, while 
still more complicated rearrangements and dismem- 
berments even to the dissolution of the molecule, may 
occur. 

The structural residue which remains after glow- 
ing the natural fluorine-containing cerium didymium 
lanthanum calcium carbonate (so-called parisite) is, 
according to G. Aminoff, surprisingly well preserved, 
so that it responds tolerably well to the searching test 
of the Laue diagram. Figs. 165-166 give a diagram- 
matic representation of this interesting case. 

A more or less extensive dehydration may be 
effected for minerals of the zeolite group without the 
destruction of the crystal form. It is known that 


Fic. 165. Fic. 166. 


Fics. 165, 166.—Laue diagrams of parisite (synchysite) and metaparisite 
After G. Aminoff 


2.01) 


Bic. 


167.—Onptics of heulandite and metaheulandite 


eS eee Sc Te RR ce ; vs 
oats SPT Mee a 
UUvERSITY ge HLS © i 


tn ‘ 


“1 


CHEMICAL ACTIONS ON CRYSTALS 145 


in heulandite, for example, the loss of water takes 
place practically continuously, and that sometimes 
an equilibrium between the hydrosilicate and its sur- 
roundings is established. Interesting parallel pheno- 
mena to this are the optical relations of the mineral. 
Like the hands on a clock, the extinction directions 
move round indicating the water content of the sub- 
stance. Here then is a good opportunity to study a 
chemical equilibrium by an optical method (e.g. by 
polarised light). Observations by O. Weigel and 
K. H. Scheumann confirm quantitatively that there 
is, in fact, a very exact parallelism between the 
chemical composition and the optics of the material, 
for their variations accord precisely. The crystalline 
nature of heulandite remains undisturbed for this 
variation to and fro of the chemical composition, at 
least as regards the first stages of dehydration, as is 
proved by the Laue diagrams. The continuation of 
the process leads to more drastic deformations. In 
the removal of water from this zeolite, so long as 
about three mols remain, we are concerned, not 
with a process very vital to the architectural sta- 
bility, but rather with the removal of a constituent 
which is only loosely, although regularly coupled 
to the silicate space-lattice. This is in complete 
agreement with the view that heulandite is mor- 
phologically closely allied to its felspar anhydride. 
The diagrams of Fig. 170 show the marked analogy 
in the appropriate angle relations. The magni- 
tudes of the axial ratios indicate the same thing ; 
but, in addition, the morphological influence of the 
H,O becomes apparent in the length of the “db” 
axis. 


In this particular we find a considerable mor- 
IO 


146 CRYSTALS AND MATTER 


— ee eee 


114"}0"' 


ee ow) a oe we es) on ee inane) Sp) Oe eee 


Fic. 169. 


Fics, 168 and 169.—Laue diagrams of heulandite and metaheulandite. 


CHEMICAL ACTIONS ON CRYSTALS 147 


phological difference in the structures of felspar and 
heulandite :-— 
Sanidine i Re eM es Gee a eae it arta halig Meek ay Rie 
Heulandite GC AON es Te 2) Ore Ba Od 


Among zeolites scolecite is of very great interest. 
Its change into metascolecite leads, just as in a 
change of modification, to an actual transformation 
of the silicate structure, indicating that here the 
water content is certainly of fundamental importance 
to the whole. Although we may use the methods 


Heulandite 


Sanidine a 
id along the (oro! . 


along the (o10) 


Fic. 170.—Comparison of sanidine and Fic. 171.—Laue diagram of 
heulandite. metascolecite. 


of ordinary and X-ray optics, which prove the 
persistence of a space-lattice arrangement in meta- 
scolecite, the phenomena may also be conveniently 
demonstrated pyroelectrically. With reference to 
Fig. 172, I found, in fact, that on dehydration of 
scolecite the front and side surfaces become inter- 
changed as regards the symmetry relations. I found 
a case of structural undermining of exceptional 
chemical simplicity in brucite, the natural trigonal 
Mg(OH),. At about 400° expulsion of the water 
begins, which wanders out from the point system 


148 CRYSTALS AND MATTER 


by diffusion and evaporates from the crystal to the 
outside, the effect increasing with increasing tem- 
perature, till finally MgO as a pseudomorph of 
Mg(OH), is left. A comparison shows that the 
crystal optics of a trigonal body, although weakened, 


Fic. 172.—-Pyroelectric effect for scolecite and metascolecite (sprinkled with 
sulphur and red lead). 

still remain with reversal of the double refraction, 

the directions of the optical axes being unchanged 

(Fig. 173). The more sensitive X-ray tests show, 

on the other hand, that the change has not occurred 

without deformation. While brucite gives a Laue 


Fic. 173.—Optics of brucite (MgO . H,O) and metabrucite (MgO). 


diagram of mere points, I found for MgO, obtained 
from brucite, a star-like X-ray figure (Fig. 174-5), a 
fact also mentioned by G. Aminoff. It indicates 
regular bending such as can be produced with the 
same effect on mica, rock-salt, and other substances. 
In addition, inner variations of the structure occur 


CHEMICAL ACTIONS ON CRYSTALS 149 


on the expulsion of the H,O. The high tempera- 
ture appropriate to the undermining is favourable 
to this. O. Paul informs me that he actually 
obtained with glowed metabrucite the Debye- 
Scherrer diagram of periclase, that is, of the iso- 
metric form of MgO, which according to him and 
Gerlach is given on heating magnesite, MgCO,, to a 
red heat. 

Such phenomena are transitional to those for 
which very drastic rearrangements in the fine- 


Fic. 174. Fic. 175. 


Fics. 174-175.—Laue diagrams of brucite and metabrucite. (X-ray star figure.) 


structure occur in the topochemical reaction of dehy- 
dration, as in the change of gypsum CaSO,2H,0O to 
the so-called subhydrate CaSO,4H,O, and then to 
the anhydride CaSO,. The constitutional difference 
between anhydrous and dihydrol calcium sulphate 
is at once indicated by the macrostereochemistry of 
the crystal form. Gypsum is monoclinic, while anhy- 
drite is rhombic, which is quite a different type of 
structure (Figs. 176-177). 

This variation in the function of water in crystal 
structure is also shown in physical chemical diagrams, 


150 CRYSTALS AND MATTER 


such as are depicted in Fig. 178. In hydrated barium 
chloride the H,O is an essential fine-structural con- 
stituent. On heating the crystal it 1s expelled in 
quanta (to some extent in large fine-structural, and 


200 


(60 


! 
| 
| 
i 
: | 


i 
Heulandite | 
‘ Ca Silz Digs On 7 7a5.5/1,0 

l 


| 
! 
| 
‘ 
| 
| 


7emperature 


| 
| | 
| | 
{ 

‘ Mols N;O—» ” “ 


Brucite 
Mg 0-120 


Fic. 176. 


a 
Fic, 177. reosrernt kl 


Fics. 176, 177.—Gypsum Fic. 178.—Diagrams for the dehydration of 
(CaSo,2H,O) and anhydrite heulandite (after O. Weigel), brucite (after O. 
(CaSo,). Westphal), and hydrated barium chloride. 


Temperature 


Ea LIne 


Dihydrated barium chloride 


Jemperature 


thus chemical, aggregates) at ‘boiling points ”’ 
corresponding to the bends in the curve of Fig. 178. 
Brucite shows one segment corresponding to evapora- 
tion, and the researches of O. Weigel and K. H. 
Scheumann indicate that for heulandite the curve is, 


CHEMICAL ACTIONS ON CRYSTALS 151 


at least to begin with, nearly a straight ascending line, 
which, according to O. Weigel, is of interest in that it 
shows singular points for the simple stoichiometrical 
ratios of silicate and water. At these positions evapo- 
ration is checked by momentary strengthening of the 
bonds between water and silicate. 

The undermining of crystals may, however, take 
place to an extent much greater than is represented 
by the removal of a relatively small part of the con- 
stituents which water usually represents. The calcium 
aluminium hydrosilicate CaOAl,036S10,¢ . 5°5 aq. of 
heulandite, for example, may be reduced fine-structur- 


Fic. 179.—a-6—Form and optics of heulandite and its silicon dioxide; 
c-d—form and optics of desmine and its silicon dioxide. 
ally to SiO... The entire filling of basic constituents 
is then removed, the result being just as though the 
skeleton of some silicious plant had been prepared by 
burning the organic wrapping. The relict of the zeo- 
lite so obtained still shows (especially after glowing, 
probably under the influence of collective crystal- 
lisation) definite optical agreement with the original 
substance, the hard rigid pseudomorph of which it 
represents. If, starting from desmine zeolite, all 
the basic constituents are simultaneously withdrawn 
with hydrochloric acid, a SiO, optically analogous 
to the original desmine is obtained. Thus the same 
chemical substance SiO, appears here to have a 


152 CRYSTALS AND MATTER 


varying structure depending on its previous history ; 
in the one case it is a heulandite, in the other a 
desmine residue. 

We may experiment in the same way with dark 
mica (biotite) and break it down to a very soft SiO, 
in flakes, similar to biotite, which, as Si0O,-metabio- 
tite, is similar optically to mica. ‘The X-ray experi- 
ments show, however, that in such extensive under- 
mining of the structure considerable disturbances in 
a leptonic sense have occurred. X-ray diagrams are 
no longer obtainable for the residual silica of the 
zeolites and mica. This is also the case for so-called 
koenenite a 3MgO. Al,O; . 2MgCl, .6H,O, which may 
be reduced to Al,Os, in very soft flakes corresponding 
to the form of the original crystal. Doubtless, in 
such cases, there occurs extensive devastation of the 
inner architecture within the external frame of the 
structure, which still stands. Using ordinary light, 
this view is not supported to the same extent as 
with the sensitive X-rays, which fail to give regular 
reflexion because of the increased agitation of the 
atoms in the structure, following the rise in tem- 
perature.t Moreover, weak double refraction may 
accompany needle, flake, or prism structure. In 
other cases, however, the weakening of the structure 
makes itself, ultimately, macroscopically evident. 
For scolecite and olivine, for example, one obtains as 
a residue of the chemical action a silica gel no longer 
coherent. The loosening of the structure has then 


1A roughness of the surface reduces the reflexion of ordinary 
light. The work of E. Wagner shows that in the same way the 
capacity to reflect X-rays is diminished by the deviation from 
planeness which arises in the planes of the space-lattice, owing to 
the increased motion of the particles following a rise in temperature. 


CHEMICAL ACTIONS ON CRYSTALS 153 


become so great that a spontaneous disintegration 
ensues. Here then is an interesting series of sub- 
stances which in their general construction and 
fine-structure form bridges between crystalline, 
amorphous solid, and, finally, on the attainment of 
the greatest dispersion, fluid materials. 

A crystallographic reconstruction corresponding 
to a substitution in molecular chemistry may be 


ee, 


Pseudomorph of SiO, Silica gel from 
Srom desmine scolecite 


Cie gi 
23. e) ae: ‘ 
aietae 
; 


oe 


Fic. 180.—(Left-hand figure) : Desmine and its silicon dioxide. (Right-hand 
figure) : Scolecite and its silicon dioxide. 


P><P<P> 


Fic. 181.—Reconstruction of chabasite: 1, chabasite; 2, metachabasite with 
carbon disulphide : 3, metachabasite with ethyl alcohol. 


easily effected for zeolites, either by replacing water 
or by exchanging, more or less extensively, the Ca 
for Na by the action of a Na-salt solution. When 
water has been removed, other substances, such as 
carbon bisulphide, alcohol, etc., may also be intro- 
duced. In every case the crystalline nature remains 
intact and specific optical characteristics are un- 
changed. 

That substances so very different from H,O 
chemically, occur as substituents in the space-lattice 


154 CRYSTALS AND MATTER 


must, however, not be assumed. Here also such ideal 
cases will be passed over as stratifications of the type 
of macroscopic intergrowths as are found so often 
in minerals; a similar arrangement may occur in 
fine-structural dimensions. It appears to me that 
the oxidation of graphite to graphitic acid is of this 
type, the latter substance showing optical properties 
(uniaxial) which graphite would show if it were 
transparent. 


RESISTANCE TO MECHANICAL DISRUPTION AND 
CHEMICAL ATTACK 


We are led from the foregoing to the view that, 
in chemico-anatomical preparation and substitution 
processes! (as have been mentioned above in a series 


Fic. 182.—Undermining during bleaching (bauerite process) and chlorite 
reconstruction of biotite: a, biotite, fresh; 4, in bleaching ; ¢, in chlorite 
process. 


of examples), there exists a correspondence with 
reactions, especially those of undermining and recon- 
struction of molecules, with which the chemist is con- 
cerned, and which, particularly in organic chemistry, 
he has so much under control. The resistances, too, 


1These may be increased by turning to processes in nature. 
The well-known bleaching (bauerite process) and very extensive 
chloritisation of dark mica are examples (Fig. 182). 


CHEMICAL ACTIONS ON CRYSTALS 155 


which occasionally oppose the transformation he 
wishes to effect, have their counterparts in crys- 
talline fine-structure. Leptonically considered, close- 
built arrangements in which the neighbouring particles 
to some extent screen one another, tend to oppose 
chemical just as they oppose mechanical attack. 
This is shown, for example, in the strong chemical 
resistance of the leptonically close-built, hard dia- 
mond, contrasting with the oxidation of the more 


Graphite 


0 


Fic. 183. —Stereograms of the closely packed, hard, chemically resistive diamond, 
and the loosely built, soft, chemically more easily attackable graphite. 


loosely-built graphite. In the same way, zeolites 
such as desmine and heulandite, with structures 
much extended by the large water content, are 
attacked by acids very much more readily than 
are their anhydrous analogues, the felspars, which 
are specifically heavier and harder. 

We thus have a noteworthy connection between 
the mechanical hardness, especially as resistance 
to fracture, and the chemical reactivity of crystals. 
It is certainly not merely by chance that gems 
such as diamond, ruby, zirconia, tourmaline, topaz, 


156 CRYSTALS AND MATTER 


rock-crystal, etc., are chemically and mechanic- 
ally strong. For these close atom packing must be 
assumed, but even then we shall not be surprised if 
very close atom arrangements go parallel with great 
softness of the material. This is the case for graphite 
(Fig. 183). The structural form indicates at once 
the origin of the very easy mechanical disrupture. 
It is due to the weak connections between the 
densely packed planes. Testing the hardness by © 
scratching separates to some extent the rigidly built, 
but loosely coupled, planes. The softness of many 
organic compounds suggests a corresponding struc- 
ture. In such compounds molecules more or less 
rigidly constructed internally will be only loosely 
linked up to one another. 

With chemical series it appears quite under- 
_standable from the fine-structural standpoint that 
forms specially stable compared with their neighbours 
should periodically arise. 

This is, indeed, a striking feature of the natural 
series of the atoms, in which the rare gases are singled 
out as terms with very stable electron distributions 
(p. 85), and which oppose, apparently with effect, 
great resistance to chemical change. Their next 
neighbours, the alkalies and halogens, on the other 
hand, exhibit the greatest readiness to react chemic- 
ally. We may add to these cases of periodically 
recurring resistance the above-mentioned breaks in 
the process of dehydration of zeolites, although here 
the effect is much less marked. Ata point of simple 
molecular ratio between silicate and water increased 
resistance is offered to the separation of the com- 
ponents. The “ lag points ’’ studied by G. Tammann 
in the structural changes of mixed crystals (such 


CHEMICAL ACTIONS ON CRYSTALS 157 


as gold-silver, gold-copper, silver chloride-sodium 
chloride, etc.), in connection with the old metallur- 
gical method of gold and silver separation by 
“ quartation,’’ are to be judged similarly, assuming 
enhanced fine-structural stability and consequent 
increased chemical resistance. According to G. Tam- 
mann, these points occur for especially simple 
distributions of the atom varieties, as for molar 
fractions 3, 4, 4, 3, etc., of the resistive component, 
which acts as a protective substance for the second 
component, which is, chemically, more easily attacked, 


XITI. AN ATTEMPT TO FORM SOME IDEA OF 
THE COURSE OF CHEMICAL REACTIONS 
FROM OBSERVATIONS ON CRYSTALS 


molecules, as well as of their fine-structural 

variations in material physical fields, is still 
in its infancy, efforts to form some idea of the 
mechanics of chemical reaction have, quite naturally, 
a merely tentative character. In order to get at the 
matter it will be advantageous to advance into this 
unknown region from various sides. We are thus 
justified, in the present undertaking, which treats 
the question from a crystallographic standpoint, in 
anticipating that the best-ordered materials are here, 
as in other cases, likely to give us useful suggestions. 


‘\ S our knowledge of the structure of atoms and 


CHEMICAL SYMMETRY ACTIONS 


The space-lattice constitution of the crystal must 
serve as the fundamental conception in this work. 
In its particular fine-structural symmetry and special 
tectonic nature are characterised the physico-chemical 
connections of the particles. 

As a result of this, for every particular case 
we find at once certain indications as to the 
mechanics of the chemical processes in the bodies 
concerned, and a basis for generalisation is obtained. 
Since, for example, CaCO, of calc-spar, which is 


constructed from Ca:: and CO’,;’ ions in a ternary 
158 


COURSE OF CHEMICAL REACTIONS 159 


rhythm (Fig. 117, p. 95), undergoes, on heating, 
the well-known reaction of splitting into CaO and 
CO., it must be assumed for the fine mechanics of 
this process that, on account of the increased heat 
motions, first the geometrico-chemical radical CO , as 
a ring of three O’s about a centre carbon, becomes 
loosened in the fine-structure. With increasing tem- 
perature these radicals, together with the calcium 
atoms, which are free moving groups in the material 
field, undergo a separation into CO, and O, which 
links up with Ca to form CaO. 

According to this, the loosening of the particles 
in the fine-structure is always to be regarded from 
the standpoint of symmetry action. Those particles 
of the structure, coupled together by rhythm or 
reflexion, participate simultaneously in the process, 
and since such coupling thus occurs throughout the 
entire crystal many million times, the process appears 
to us in analytical chemistry as a discontinuous 
change, possibly in a series of steps, if the new 
arrangement contains the departing component again 
in definite symmetry disposition, as is the case, for 
example, in the ignition of gypsum to the subhydrate. 
A practically steady variation can have its origin in 
complicated re-groupings of the point system, closely 
following on one another. 

For non-crystalline substances such as gases and 
liquids the relations in the molecule cannot, as 
regards main principles, be thought of in any other 
way. Crystal regularity is, indeed, only a special 
case of fine-structural arrangement. The four H’s 
of an individual CH, molecule are, in this sense, 
coupled up in a symmetry arrangement just as the 
three O’s in the CO, radical cale-spar. They must 


160 CRYSTALS AND MATTER 


participate simultaneously in action as markers of 
the equal valued corners of the tetrahedral molecule, 
so long as this symmetry persists. . 


PRE-CHEMICAL PROCESSES AND DISCONTINUOUS RE- 
ACTIONS. MASS ACTION AND CATALYSTS. HEAT 
AS AICATALYST, 


In following up the above observations, a very 
important point must be discussed. Since it actu- 
ally happens that in the chemical field, i.e. in the 
reciprocal action of several types of molecule, one 
only of the four H’s of the CH, molecule may 
participate in chemical action (say CH, + Cl, = 
CH;Cl + HCl), then on the basis of the symmetry 
action set forth above, it is necessary to assume that 
the four H’s of CH,, before the completion of the 
chemical reaction, that is, in a pre-chemical process, 
will be differentiated by the fine-structural pro- 
minence of one of their number. The tetrahedral 
placing of the four H’s must have become changed 
under the reciprocal anisotropic influence of neigh- 
bouring molecules CH, and Cl, in such a way that 
one of the four has obtained a singular position in 
the fine-structure, the other three remaining equiva- 
lent. The four H’s, instead of representing the cor- 
ners of a tetrahedron, mark out those of a trigonal 
pyramid (Fig. 184). The hydrogen at the apex is 
in a certain sense connected to the remainder of the 
molecule by very weak threads. It is these which 
naturally give way first when the mutual change of 
form of the deforming interacting molecules exceeds 
a certain measure of tension. Substitution in this 
stereochemical body occurs localised at the hydrogen 
atom, which has become particularised in the fine- 


COURSE OF CHEMICAL REACTIONS 161 


structure, and a new stable arrangement is set up. 
When, in the case of more complicated substances, 
splitting occurs, depending, of course, on the appro- 
priate molecular structure, the process is directly com- 
parable with the rupture of the internal connections 
of crystals during cleavage which cuts through the 
weaker bonds. In aliphatic compounds C — C coup- 
lings are, according to Wollers, weak arrangements ; 
for aromatic compounds separation occurs more 
readily between C and H. 

The law of mass action is, in the above sense, the: 
expression of the fact that numerous deformation 
forces keep the fine-structural displacement con- 
stantly directed towards one side. 

In addition, the analogous role of catalysts in the 
fine-structure becomes evident. The tension neces- 
sarily preceding chemical action may well be increased 
by the presence of a third type of molecule. The 
action may, in some cases, be initiated by such a 
third party. In the actual chemical transformation 
the auxiliary substance does not participate, and, 
in consequence, suffers no loss: it can officiate in 
innumerable cases, one after the other, in the mole- 
cular swarm, and in so doing produces a great effect, 
although present in very small quantity. Thus such 
material catalytic factors function pre-chemically. 
The substances concerned represent catalysts as they 
deform the fine-structure. If it be desired to include 
the preparatory tension process in the chemical 
action, there is no formal objection. Physical and 
chemical processes merge into one. I think the 
observations of J. Stark and myself are pertinent in 
this connection. 


Raising the temperature, as the acceleration of the 
IJ 


162 CRYSTALS AND MATTER 


internal fine-structural motions, can be similarly 
considered as catalytic. It is understood that in this 
case, weak bonds between structural groups will give 
way sooner than they would at lower temperatures ; 
they are, we may say, pulled about at the higher 
temperature. If, for example, in NaCl. 2H,O, the 
water molecules, which do not concern the mono- 
valency of Na and Cl, and are but loosely held by 
co-ordination bonds, the thermal oscillation of the » 


CMG 2, = GAG *H0 ee C72 
a 


My *Cl, 2 CHC? +HCL ” oe Ch; 
Af 


at 
CH, CH, H Chg? 
‘ 44 
KT 
H 

Vy 4 a 

“4 cd 

Fic. 184.—Fine-structural schemes for the action of chlorine on benzene and 
methane respectively. 


particles is considerably increased, the weak binding 
forces will be overcome first. Water is suddenly 
liberated in cases where the molecules are dissimilarly 
attacked, as for BaCl,.2H,O, in two stages, one 
after:the other {at 105° dnd 162°. see Digmergs 
p. 150) ; sometimes in even more, as for CuSO, . 5H,O. 
When the tension becomes sufficiently large under 
the influence of the rise in temperature, “ valency 
tensors’ also break apart such as those between the 
ions Ca and CO; during lime-burning and in other 
similar cases of chemical decomposition. 


COURSE OF CHEMICAL REACTIONS 163 


To illustrate these points the schemes of Fig. 184 
are shown, which refer to methane and benzene as 
typical cases. The final structure there derived for 
C,H;Cl appears to me to agree completely with the 
diagram already published by J. Stark in his excellent 
book, “ Die Elektrizitat im chemischen Atom,” a 
happy case of the agreement of results derived from 
different standpoints. 

It was also of interest to me, on looking through 
the literature, to learn from a hint by E. Farber 
in “‘ Naturwissenschaften,’ that, in the delibera- 
tions of the older chemical generation, representation 
of a weakening of the bond in the molecule before 
the occurrence of the chemical reaction occasionally 
played a role. This is seen in the assertion of 
A. Kekule, who says that “ during the approach of 
the molecules the connections of the atoms in the 
same are already weakened, for one part of the 
chemical affinity is bound by the atoms of the other 
molecule until finally the previously united atoms 
entirely lose their interconnection and the newly 
formed molecules separate.”’ 

One ventures to extend the scheme in the above 
to the assumption of a pre-chemical molecular 
deformation. 


CRYSTALLOGRAPHIC INDICATORS OF CHEMICAL 
PROCESSES 


Since the physical, chemical, and _ crystallo- 
graphic considerations agree, as they do, we are 
now in the position to corroborate, to some extent 
at least, the assumption of preliminary structural 
changes in crystallographic experiments. In par- 
ticular, observation of the conditions for certain 


164 CRYSTALS AND MATTER 


crystallisations lead once more to the postulation of 
molecular fore-forms in solution from which crystals 
are separating, an assumption which has already been 
mentioned on page 41. While, for example, CaCO, 
salt separates out from a pure calcium carbonate 
solution in trigonal form of the 3m class as calc- 


Fic. 185.—Calc-spar and aragonite. 


spar, experiment shows that the addition of mag- 
nesium sulphate to the solution causes the forma- 
tion of a stereo-chemically different variety, digonal 
aragonite of the 2m group. Thus one or other 
modification of CaCO; must certainly be predeter- 
mined by molecular pre-forms in the solution. A still 
more varied example 
of this has been in- 
vestigated by O. Pauli 
in my institute; his 
experiment deals with 
acid phenyl acridon- 
Fic. 186,--Monodiinicand triclinic modifies- 14m sulphate as tis 
tions of acid phenyl acridonium sulphate. appears in different 
modifications according to the proportions of water, 
sulphuric acid, and alcohol in the solution. Figs. 186 


1 Probably as CaCO; or CaOCO,, possibly as a loose compound 
with MgSQ,. 


COURSE OF CHEMICAL REACTIONS 165 


and 187 give diagrammatically the appropriate} 
conditions. 
For the most part, then, we are supported in 
the conclusion that chemical reactions do not occur 
abruptly, but after preliminary actions, in cases 
which permit a leisurely although indirect observation 
of the changes of state by means of physical indi- 
cators. Occasionally that is the case for changes of 
crystallographic modifications, which are not, indeed, 


Fic, 187,—Crystallization diagram for acid phenyl acridonium sulphate. 


merely physical, but also chemical actions (p. 70). 
For the investigation of the general course of the 
fine-structural processes inside the substance, optical 
methods may be used as in the elegant studies of 
A. Hantzsch and his pupils, where absorption 
phenomena in the ultra-violet were employed as 

1 Different molecular pre-forms of crystallisation will arise if at 
higher temperatures, or with certain other substances in the solution, 
salts poor in water crystallise out. The same holds good if at low 
temperatures, or in the presence of other substances, salts rich in 


water are formed. The diagrams of van’t Hoff and D’Ans, in par- 
ticular, furnish classical examples of this. 


166 CRYSTALS AND MATTER 


indicators of chemical processes. For the stereo- 
chemical changes to be determined here, investiga- 
tion of the refractive index is helpful. In this 
particular I have studied exactly, with R. Kolb, 


15850 


* 700° -s9° * 50° 100° 150° 200° o ° o * 450° ° * 650° a o oe 
ST 8 oe i Oo” 250 300 540 400° 450° 500° 550) 600° 650° 700° 750° 800 


Fic. 188.—Curves of the refractive indices w of 8 and a quartz for various kinds 
of light. 


S17 OF. 200° 400° 600° 800? 


Fic. 189,—Curves of the refractive indices w and « of 8 and a quartz for sodium 
light characterising the variation of the double refraction. 


such a physico-chemical process for quartz, which, 
on exceeding 575°, changes from the trigonal f into 
the hexagonal a state (3827 6s) (Fig. 88, p. 71), as 


the Laue diagrams show in very neat fashion (Fig. 89, 
-p. 71). With respect to the refractive indices, 


COURSE OF CHEMICAL REACTIONS 167 


Figs. 188 and 189 explain fully. It is clearly seen for 
the case in question, 8 +a quartz, that on nearing 
575. the gradient of the curves is much increased, 
and at the temperature named exhibits a discon- 


FIG. 190.—Curve of the angle variation of 8 and a quartz. 


tinuous drop. This line must be regarded as a 
definite indication that the process 3s ~ 6s quartz 
is led up to by a gradually developing tension 
in the structure; this increases as a pre-chemical 
action until the sudden 
rearrangement by a dis- 
continuous change in the 
fine-structure. 

Similar conclusions to 
those obtained above fol- ene ear 2 pieeners variation 
low from the observations 
which I carried out in collaboration with R. Kolb 
on quartz with respect to its morphological variation 
on transformation (Fig. 190). F.E. Wright obtained 
similar results for the same mineral. 

The thermo-goniometrical researches of R. Gross- 
man made, under the direction of P. Niggliand myself, 


Se esks, 


400 600 


S 


200° 


168 CRYSTALS AND MATTER 


on borazite and leucite (Fig. 191), show analogous 
results. 

Although the fine-structural relations of quartz 
are not determined experimentally, still, by resorting 
to the representations of J. Beckenkamp, and 
especially by following the discussions of P. Niggli, 
it is possible to make a provisional diagram for the 
transition of the quartz modifications. In Figs. 192 
and 193 such a scheme is depicted. The arrows in © 
the diagram of the screw trigyric structure indicate 


© 
Hexagonal Quartz 


Fics. 192 and 193.—Fine-structural diagram symbolising the transition 
3s => 6s of the quartz modifications. 


O 
the tendency of the O particles in the SiC ii triangle 


to set themselves in the screw hexagonal arrange- 
ment, a tendency which steadily becomes more - 
effective as the temperature rises; finally, the 
sudden rupture of the tension which has increased 
to the limit gives the 8 form. 

Thus crystallographic considerations support the 
assumption that one can, in ideal schematic fashion, | 
represent a chemical transformation as the action of 
a physical or chemical field of such a character that 
the change in the chemical structure is led up to by 
a state of fine-structural strain and deformation ; 


COURSE OF CHEMICAL REACTIONS 169 


this becomes increasingly pronounced, and finally 
leading, by a sudden adjustment, to the new stable 
system. 

Many gradations of the relation may, of course, 
arise. In particular, owing to great resistance, the 
period of strain may be more or less diminished so 
that the discontinuous chemical action then occurs 
almost or entirely without this intermediary state. 
On the other hand, a pre-chemical deformation may 
become very much extended, so that the discon- 
tinuous change is correspondingly lessened or absent 
altogether. The transition of modification, from 
aragonite to calc-spar by heat, points to this. With 
a view to learning more about the change, I suggested 
to K. Wiinscher a thermo-goniometrical and thermo- 
optical research. It was shown that angle and 
refractive index variations, on heating the mineral 
to 325°, are a function of the temperature, increasing 
and decreasing with it. For higher temperatures, 
however, the tension process in connection with the 
transformation of aragonite into calc-spar occurs to 
some extent in spontaneous glidings, for at constant 
temperature the form and optical properties of the 
mineral vary, the variations being more rapid the 
higher the temperature taken. Finally, the sub- 
stance completes the transition of one into the other 
modification by a sudden adjustment. 


XIV. ANALOGY OF THE MORPHOLOGICAL 
ACTION OF PHYSICAL AND CHEMICAL 
FIELDS ON CRYSTALS 


T is of considerable interest to compare the . 
[ ecsis observable homogeneous deformations of 

the crystal structure which occur under the in- 
fluence of heat, with variations of crystallographic 
form in the chemical field. 


THERMAL INFLUENCES ON THE CRYSTAL FORM 


The action of temperature change on the mor- 
phology of the crystal becomes apparent, as is well 
known, in explicit formal symmetry actions, the 
general type of which is most easily studied for 
spheres. Such forms remain, for uniform rise of 
temperature, isoradial if they are composed of iso- 
metric substances. The sphere remains for tem- 
perature variations as such intact. The change is 
restricted to an alteration of the radius. Crystallo- 
graphic ternary, tetrad, or senary substances, on the 
other hand, show transformation of the initial form 
to a rotation ellipsoid, the axes of rotation coinciding 
with the crystallographic main-axes. Spheres of 
rhombic, monoclinic, and triclinic substances finally 
give rise to triaxial ellipsoids, the principal direc- 
tions of which are arranged in accordance with the 
symmetry (Figs. 194-106). 

Although the morphological reaction to change 


in the heat motion of the particles appears usually 
170 


MORPHOLOGICAL ACTION 171 


simple, the complicated interlacing of fine-structural 
force fields is shown here in the occasional contrac- 
tion and not extension, with increase of temperature. 
It may, in fact, happen that for anisotropic sub- 


Fic. 196. FIG. 197. 


Fics. 194-196.—Schemes for the homogereous deformation of isometric, 
uniaxial and trimetric crystals on heating (initial sphere shaded). 


Fic. 197..-Scheme for the homogeneous deformation of a calc-spar sphere 
on heating. Coefficient of expansion in the direction a@ = 0:0,2621, 
¢ = — 0:0,0540 in the intermediate direction shown = 0 (initial sphere 
shaded). 


stances, in certain directions, dilatation occurs, and 
in others contraction. For cuprite it happens that 
with rise of temperature in the region below — 4:3°, 
isoradial contraction takes place. On heating tri- 
gonal calc-spar it expands along the rotation axis, at 


172 CRYSTALS AND MATTER 


the same time contracting in all directions perpen- 
dicular thereto. As a result, radii of the initial 
sphere, inclined at 65° 49:5’ to the principal axis, 
will not be altered in length by temperature change 
(Fig. 197). 

Hexagonal silver iodide has, on the other hand, 
a negative expansion coefficient along the crystallo- 
graphic vertical (a, = — 0:0,0397) ; in the horizontal 
direction it expands on heating (a, = 0°0,065). The » 


Fic. 198. Fic. 199. 


Fics. 198 and 199.—Schemes for fine-structural variation in the physical field. 
Preservation of the indices and zone relations. 


cubic coefficient a, + 2a, 18 = — 00,0267. The 
volume of the salt is therefore reduced by rise of 
temperature. 


From a fine-structural aspect such relations may 
readily be understood in the symbolical representa- 
tion, as an alteration of the distance between the 
centres of heat motion corresponding to the visible 
change of form. The Figs. 198 and 199 above show 
this with the necessary diagrammatic exaggeration. 

In these figures one recognises as the ruling con- 


MORPHOLOGICAL ACTION 173 


ditions the preservation of the symmetry, of the 
parallel edges (the zone relations), and of the indices 
which, in the triangular surface shown retain their 
unit values Ia: 1b: 1c. Angles and axial ratios alter 
within the limits of the prevailing symmetry. 

The extension coefficients, the order of which has 
already been given in the special case of Fig. 197 
give us an idea of the absolute value of the varia- 
tions. They are usually very small. For example, 
in the elementary cube of rock-salt (Fig. 24), p. 22), 


200° -100° 0° ~~~-*400® ~~ *200° ~~ +300" +400" —+500° OOF *700° 
FIG. 200,—Variation of the cleavage angle of the plagioclases, albite, labradorite, 
and anorthite. 

the side length increases merely from 5-63 x Io78 
em, at 0° to 5:77 x 107* cm.at 500°. The angular 
deformations of anisotropic substances known since 
the time of Mitscherlich (1799-1863) are correspond- 
ingly small. 

To extend his studies on calc-spar I investigated 
the angle variations of the rhombohedral cleavage 
form of this mineral over the extensive tempera- 
ture interval from — 165° C. to 596° C., i.e. for 
761°. I found a change of angle of about 1° 9’ 20”, 
i.e. about g:1’ to each 100° on an almost linear 


174 CRYSTALS AND MATTER 


graph. Usually the thermo-morphological reaction 
is even smaller. For quartz I measured a change 
of the rhombohedron angle of only 14:0’ for 553° 
(20° — 573°), 1e. not 3’ per 100° C. 

Of course, in such variations of form sometimes 
there arise very complicated fine-structural processes, 
which may be inferred from appropriate diagrams. 
Although the curve obtained in the example of calc- 
spar rises almost linearly, in other cases well-marked ' 
curvature is shown as for quartz, already referred to 
(Fig. 190), and the plagioclases. In Fig. 200 the 
second curve, which relates to an isomorphous mix- 
ture of albite and anorthite, termed labradorite, also 
deviates from the arithmetic mean of the other two, 
showing that the angle and the accompanying fine- 
structural variations in such cases are not. simply 
additive. 


CHEMICAL INFLUENCES ON THE CRYSTAL FORM 


The analogous inquiry as to the morphological 
action of a chemical field on the fine-structure fails in 
general owing to the impossibility of magnifying the 
effect sufficiently for observation. If, for example, 
a crystal of calc-spar is suspended in water, then a 
deformation of the crystal is certainly to be assumed, 
but cannot be rendered visible experimentally. In 
contrast to the thermal action, the influence of the 
chemical field is restricted in the above case to the 
surface. It can make itself felt in the peripheral 
processes of growth and solution of the crystal, but 
not markedly in transformation of the structure. 
Moreover, a simultaneous action throughout the 
whole body of the crystal, i.e. permeation of the 
liquid to all parts, would be necessary to correspond 


MORPHOLOGICAL ACTION 175 


to the effect of heat. For some crystals that is 
actually brought about, as for those of albumen. 
Indeed, the process exceeds in definiteness all expec- 
tations. Albumen crystals take up water either from 
the surrounding liquid or from an atmosphere con- 
taining water vapour. In this case, then, the par- 
ticles of the space-lattice are surrounded with water 
molecules. In the reciprocal anisotropic action be- 
tween the crystal structural groups and the water 
particles regularly arranged about them, an unusually 
large deformation of the crystal makes its appearance, 
the crystallographic symmetry remaining unaltered. 


Albumen crystal 


Before imbibition ; Af ter t 
imbibition 


Fic. 201.—Homogeneous deformation of a crystal cf albumen by imbibition. 


The isometric albumen crystals swell up, remain- 
ing trigonal, with anisotropic variation of the angles. 
According to A. F. W. Schimper, to whom chiefly we 
are indebted for the appropriate observations, the 
plane polar-angle of trigonal albumen (obtained from 
Brazil nuts) changes from 604° to 39$°, and thus by a 
very largeamount. Fig. 201 shows diagrammatically 
a similar case of extensive deformation for the cubic 
rhombohedra of albumen occurring in solanine. 
These are drawn out to a very definitely acute- 
angled form, the plane polar angle of which amounts 
now to 68° instead of g0$°. It is especially interesting 
that for albumen from Brazil nuts, perpendicular to 
the ternary axis no observable swelling occurs, whilst 


176 CRYSTALS AND MATTER 


the linear measure of the enlargement in the axial 
direction is very considerable. 

The optics of albumen crystals deformed by 
swelling in water show a regular variation, as is also 
the case for those expanded thermally. Isotropic 
crystals remain, with variations of the refractive 
index, isotropic; double refracting forms alter the 
magnitude of this property. All return again on 
evaporation of the water to their original states.1, 
According to this, the morphological actions of 
thermal and chemical fields in crystals are exactly 
similar, probably an indication that the thermal 
process also is to be regarded from a chemical stand- 
point, and, as a bombardment of the structural 
particles by electrons, is analogous to chemical 
action. 


COMPARISON OF THE THERMAL AND CHEMICAL 
INFLUENCES ON THE CRYSTAL FORM 

A comparison of the fine-structural effects follow- 
ing temperature variation, on the one hand, and 
under the influence of the chemical field, on the 
other, may best be carried out with respect to iso- 
morphous substances. In this connection the follow- 
ing table (p. 177) will be of interest. It indicates the 
relatively large effect of a chemical substitution of 
Cl by Br or I in the potassium halides compared with 
that of a temperature change of about 500°. The 
molar cell and molecular domains (p. 107) relate to 
cube forms. 

For KCl, then, the effect on the external form, 
measured by the axes of the molecular domain for a 
rise of temperature of about 500°, is to that produced 


1 Acids apparently break up the structure. 


MORPHOLOGICAL ACTION 177 


Mol Domain. Cell Domain, Molecular Region. 
KCl. 

Volume Axes i en eer ela ae 

tie nes crn Cm. Cc. Cm. om 
20° ; eH) 39764 3°346 247°72 6°280 61°93 3°956 
ROO | . | 40°19 3°425 205°72 6-429 66°43 4:050 
2°55 0°079 18-00 O-149 4°50 0°094 
20° KCl mh) 3704 3°346 247°72 6-280 61°93 3956 
KBr .| 43°19 37508 285°52 6°585 71°38 4°148 
5°55 o'162 37°80 0°305 9°45 0°192 
4 Wl AGtTO 37508 285°52 6°585 71°38 4°148 
Gee eh 5207 4) 3750 41) 350°24 1). FOO | 87-56. | agate 
19°78 0°248 04°72 0°464 16-18 0:293 
olen -| 37°64 3°346 247°72 6:280 61°93 3956 
Bi -| 52°97 3°756 | 350°24 | 7:049 | 87°56 | 4-441 


by a substitution of Br or I for Cl as 1: 2:04: 5:16. 
Naturally, the greater the change of temperature the 
larger will be the consequent variations, within the 
limits of one modification. In Fig. 200 of the felspars 
(p. 173), it is recognised how closely in such cases 
thermal and chemical effects can resemble each other. 

The phenomena in question possess further a 
special interest, in that the action of a rise of tem- 
perature in the crystal and the loosening effect of a 
permeating substance are gradations in the process 
of melting and solution, that is, in the process of “‘ren- 
dering amorphous.’ Considered in connection with 


the series of metamorphoses (p. 69), which substances 
12 


178 CRYSTALS AND MATTER 


pass through on raising the temperature, continuous 
thermal homogeneous deformations figure as pre- 
liminaries to the abrupt collapse of the space-lattice 
arrangement, a process which generally runs parallel 
with the external phenomenon of melting, i.e. of 
“ flowing apart.’”’ Under the influence of increased 
fine-structural agitation the crystal form is com- 
pletely broken up, in accordance with Lindemann’s 
views, if the vibrations of the particles about their, 
positions of rest become commensurable with their 
distance apart, and so lead to their collision. The 
domains of the fine-structural groups merge together, 
and the forces binding the lattice are overcome 
by the disruptive action of the heat motions, the 
crystal form is destroyed and scattered into irregu- 
larly placed new kinetic units. In this sense the 
fine-structural deformation which precedes the sudden 
change appears analogous to the pre-chemical pro- 
cesses referred to previously (p. 160). The albumen 
crystals during imbibition strive in an exactly similar 
way to become amorphous. It is of great interest, 
and also characteristic, that the structure can be 
linearly extended before dissolution to such a large 
extent, often by several times its original length. 
Even with extensive dilatation the interleptylic 
fields of force give rise to some cohesive action. 
That must be ascribed to a regular incorporation of 
the water particles, which arrange themselves analo- 
gously to the H,O in zeolites, and acting as a chemical 
cement bring about the observed cohesion. The water 
is present, however, in much greater quantity than 
in zeolites. Finally, there occurs separation into 
irregularly arranged particles, as may be observed 
very neatly, according to A. F. W. Schimper, in 


MORPHOLOGICAL ACTION 179 


small crystals of albumen from the seeds of the 
castor oil plant. These, being cubic, swell isoradially 
in dilute sulphuric acid to three or four times their 
Original diameter, and then immediately go into 
solution. To what extent, in structures of so many 
atoms the space-lattice arrangement is lost on tran- 
sition to the colloidal, and finally to the molecularly, 
disperse state, further X-ray studies must show 
(compare p. 65). 


XV.CRYSTAL PHYSIOLOGY: ANDi tis 
CLASSIFICATION OF ATOMS 


THE STANDARDS AND PHYSIOLOGY OF THE 
CLASSIFICATION 


CIENTIFIC classification, as a concise charac- 
G essation of the peculiarities and relations of 

the objects investigated, is of considerable 
importance. Its development must move parallel 
with the advance of knowledge, conforming to the 
broader purpose of ensuring simple and natural 
methods in our deliberations. In these times of 
radical changes in our ideas of the nature of matter, 
the systematic co-ordination of the results of investi- 
gations merits careful attention. The importance of 
the classification as the characterisation of the fine- 
structural particles and their family relationships is 
supported in a gratifying fashion by independent 
lines of thought from many directions. Our ideas 
of the constitution of atoms as neutral and ionised 
forms, both of normal weight and as _ isotopes, 
together with the conception of the element, play 
the leading role here. It appears to me not inap- 
propriate in these questions also to emphasise the 
close connection of the various states which runs 
through the fine-structural series, electrons, atoms, 
ions, molecules, and crystals. The crystal, the 
highest and especially regular member of the series, 


easily observable in its external form and physical 
180 


THE CLASSIFICATION OF ATOMS 181 


conditions, enables the general idea of the principles 
of classification to be readily grasped. In particular, 
it is clearly seen that considerable physiological 
breadth of property is to be ascribed to a leptonic 
unit. The appearance of a rigid regular form and of 
an inner homogeneity, say for ruby, is an illusion. 
A change of temperature changes the volume of the 
crystal, and in the case mentioned its form also. 
Optical tests of the refraction, double refraction, and 
absorption show that this crystalline form can 
experience changes in its inner constitution which 
are to be traced back, finally, to reversible rearrange- 
ments of the fine-structure. 

X-ray data testify that, in correspondence with 
this general conclusion, the motion of the fine- 
structural particles is highly variable. Even analy- 
tical differences, as in the case of isomorphous 
mixtures, with its powerful influence on the optical 
absorption and the specific gravity, or the entrance 
and exit of water which occurs for zeolites, can arise 
without prejudicing the idea that we still are dealing 
with the same kind of crystal whose physiology alone 
changes within certain limits. In systematic classifi- 
cation the type retains its place, despite these varia- 
tions. It seems to me that the transfer of such views 
to the classification of leptonic forms leads to a simple 
and natural formulation. 


ELECTRONS, ATOMS, AND MOLECULES 


The fundamental constitution of all things lies 
hidden in the electron as the elementary quantum of 
electricity and the primary constituent of matter ;1 


1If they have not to relinquish this rank to the archons as 
vortex pairs as suggested by O. Wiener. 


182 CRYSTALS AND MATTER 


electrons are therefore of the first importance. 
Their division into e+ and e~ is, however, of so great 
systematic simplicity that it has, up to the present, 
sufficed. 

The case is very different for the atoms, the 
classification of which has developed into a special 
study. Their general characteristic in the manifold 
of forms lies in the presence of a nucleus within the 
structural unit. | 

The highest grade of individual leptonic structures 
is represented by the molecule. Its special feature is 
that of a combination of atoms to a new unit, thus 
the presence of more than one nucleus in the kinetic 
unit. 

Everything else in fine-structural phenomena, as 
they are presented in the gaseous, liquid, and crystal- 
line states in quite inexhaustible abundance, comes 
under the head of modes of aggregation of the elec- 
tronic, atomic, or molecular fine-structural forms. 
The force of the classification lies in the atomistic 
structural gradations. 


ATOM TYPES 


Reviewing, therefore, the scientific facts relating 
to the atomic units, there is now no further doubt 
that these units must be arranged in the order of 
their atomic numbers (corresponding to the X-ray 
spectra, p. 20). This is done in the following table 
(p. 183). In each case the atomic weight is sub- 
joined thus, A.W., as it is not essential to the 
classification. 

The terms of this series will, in accordance with 
scientific requirements, be designated atom types, 
_and for the further systematic subdivision of these 


THE CLASSIFICATION OF ATOMS 


183 


the expression atom sub-types is introduced. The 
table indicates the special property of atom types 
—the atomic number which, in accordance with 
the nuclear charge, fixes the position in the series 


omer to Ur. 


Atomic 
Number. 


O ON OAMUAWN 4 


Name. 


Hydrogen 
Helium 
Lithium 
Beryllium 
Boron 
Carbon 
Nitrogen 
Oxygen 
Fluorine 
Neon 
Sodium 
Magnesium 
Aluminium 
Silicon . 


Phosphorous 


Sulphur 
Chlorine 
Argon 
Potassium 
Calcium 
Scandium 
Titanium 
Vanadium 
Chromium 
Manganese 
Iron 2 
Cobalt . 
Nickel . 
Copper . 
Zinc 4 
Gallium 
Germanium 
Arsenic 
Selenium 
Bromine 
Krypton 
Rubidium 
Strontium 
Yttrium 
Zirconium 
Niobium 
Molydenum 


Ruthenium 
Rhodium 
Palladium 


Atomic 
Number. 


Name. 


Silver 
Cadmium 
Indium 
Tin 
Antimony 
Tellurium 
Iodine . 
Xenon . 
Caesium 
Barium 
Lanthanum 
Cerium 


Praesodymium 
Neodymium . 


Samarium 
Europium 
Gadolinium 
Terbium 
Dysprosium 
Holmium 
Erbium 
Thullium I. 
Ytterbium 
Lutetium 
Hafnium 
Tantalum 
Tungsten 


Osmium 
Tridium 
Platinum 
Gold 
Mercury 
Thallium 
Lead 
Bismuth 
Polonium 


Emanium 
Radium 
Actinium 
Thorium 


Protactinium . 


Uranium 


The characteristic of an atom is, 


184 CRYSTALS AND MATTER 


therefore, its numbered place in the atomic series ; 


briefly, its monotopy. 
No term of the series can be dispensed with ; 


) ° i 
70oF & A 
<= t 1 
s Mgt 
sol = Rb Nias 
S fon 
2 t { 
sok = Pe 
: Be 
Ba te 
Sr : 
30 ! 
Nad 16n Br rs : ; Th 
Pf Bi 
Tl 
Hy U 
PL PAy 


Atomic weights 
200 220 


180 2 


£00 i420 0 T40 9460 
Fic. 202.—Graph of atomic volumes. 


each one fulfils the task of representing a necessary 


number in a sequence. 
The periodic table of L. Meyer and D. J. Mendeleeff 


Logarithm of the diameter of 
the crystallographic ion 
domain 


Logarithm of the atomic number 


Fic. 203.—Graph showing series of atoms. After E. Schiebold. 


accomplishes in the familiar way a grouping of the 
atom types. The curve obtained for the atomic 
volumes (Fig. 202) serves the same purpose. With 


THE CLASSIFICATION OF ATOMS 185 


reference to our ideas on crystallographic atomic 
domains, it appears of interest to indicate here a 
graphical arrangement of important atomic groups 
obtained by E. Schiebold. This is arrived at by 
taking the logarithms of both atomic number and 
the diameter of the atomic domain; the groups are 
signified by the arrangement of the points in the 
diagram which are often linear in series. 


ATOM SUB-TYPES 


The monotopy of the atom types is quite com- 
patible with physiological differences which do not 
interfere with the constancy of the nuclear charge 
which is characteristic and equal to the atomic 
number. In this way atomic sub-types may pos- 
sibly arise in the series of monotopes, characterised 
by fine-structural differences either in the region of 
the satellite electrons or in the nucleus. 

We may therefore assume within the atomic types 
the following atomic sub-types. Differences in the 
outer electron shell differentiate neutral atoms from 
atom ions, which in turn separate into cations or 
plus atoms and anions or minus atoms. Owing to 
the extreme lightness of the electrons, the theoretical 
changes of weight arising from omission of a few 
negative particles from the outer shell, or by their 
entrance, are inconsiderable ; neutral atoms are, with 
their corresponding + or — ions, practically isobaric. 
For other atomic sub-types a difference of the mass 
content, actually in the central nucleus (without 
change of its charge), of so extensive a nature occurs 
that a difference of atomic weight is observable. 
This is the case for isotopes, so termed in K. Fajan’s 
fundamental papers on the subject. The members 
of such a group are, therefore, heterobaric. 


186 CRYSTALS AND MATTER 


ELEMENTS 


Another criterion in classification is obtained from 
our conception of the element. Its characteristic 
is that its constitution consists entirely of monotopic 
atoms ; thus elements are possible containing either 
only one atom sub-type or several different varieties. 
F. Paneth expresses this very neatly by differentiating 
between pure elements and mixed elements. The 
former contain only one atom sub-type, the latter — 
more than one. 


NORMAL MIXTURE AND SEPARATION OF ISOTOPES 


Of primary importance in analytical chemistry 
is the remarkable fact that for elements containing 
heterobaric components a normal mixture is invari- 
ably found. For chlorine, with isotopes of mass 
35, 37, and 39, the ordinary atomic weight 35:46 is 
always shown (whether the chlorine is extracted 
from eruptive minerals or deposited sediment, 
whether it is of.terrestrial or meteoric origin), a cir- 
cumstance which reminds us, at least formally, of 
the equilibrium phenomena of eutectics. Thus ordi- 
nary. Brys-9,°) is. 0°46 Br,, + 0°54. Br, >) Cie 
0:23 Cl,, + 0:77 Cl,;; (neglecting the small amount 
Of .Cl,o}.3) Slpg-s ==’ 0230 [Oleg 4: 0:70.55, 5 ae 
0:97 Ayo -+ 0°03 Age, etc. 

The mixed isotopic constitution of many of the 
atom types also enables us to understand why, in 
the Mendeléeff system, there occur occasionally 
deviations from the arrangement in order of atomic 
weight. For atoms of nearly the same weight it 
may easily happen that this admixture displaces an 
atom type from the natural sequence to a false 


THE CLASSIFICATION OF ATOMS § 187 


position, as is the case for argon. If the lighter 
component (Ar;,) were present in somewhat greater 
proportion than is the case in the normal argon 
mixture (Afso.s3 = 0°97 Ar, + only 0:03 Ars,), the 
rare gas would immediately assume its correct place 
in the Mendeléeff system before potassium (A.W. 
39°I0). 

A partition of the isotopes by arrangement in 
separate positions in the fine-structure occurs in every 
crystallisation of a substance containing such atomic 
mixtures (compare Fig. 119, p. 98). As is well 
known, a separation of the components to a detectable 
extent has been accomplished in the researches of 
F. W. Aston who continued the work of Goldstein, 
W. Wien, J. J. Thomson, and others. By the 
different deviations of the ions in electrostatic and 
magnetic fields, Aston separated and identified the 
individual isotopes, one of the finest results of 
general scientific endeavour in the direction of a 
unified concept of matter as the aggregation of a 
primary constituent. The integral atomic weights 
of the iostopes point to this. Here the anomaly of 
hydrogen, with its non-integral atomic weight 1-008 
compared with oxygen = 16, is not yet explained, 
but it now merely spurs us on completely to establish 
this otherwise predominant concept by further experi- 
mental work and study. 


CONCLUSION 


In reviewing all the various experiments and 
arguments dealt with above, which in the present 
early stages of fine-structural investigation naturally 
more often pass in a mere mention than lead to 
definite results, it is recognised that crystals are 


188 CRYSTALS AND MATTER 


actually in many respects typical of the general 
conception of the constitution of matter. In their 
macroscopic form and their physico-chemical relations 
are reflected, not only the fine-structure and the 
physics and chemistry of their own particular micro- 
cosm, but also of matter in general. With their 
three-dimensional regularity and easy accessibility 
to direct observation, they are specially suited to 
serve as the starting point in the investigation of . 
laws universally valid. In this way crystallography 
stimulates the physicist, chemist, and natural phi- 
losopher, as it itself, on the other hand, has gratefully 
received so much help from these great sister sciences. 

In such common endeavour the sowing of the 
fine-structural soil cannot fail to germinate vigorously 
and, on progressive cultivation, to develop into a 
rich harvest. I hope that this present exposition 
will serve as a small contribution to the great work. 


Readers wishing to acquire a more detailed know- 
ledge of crystal science, are referred, to the following 
series of works on the subject :— 


TEXT-BOOKS ON CRYSTALLOGRAPHY 


J. BEcKENKAMP. Leitfaden der Kristallographie, 1919. 

W. H. and W. L. Bracc. X-Rays and Crystal Structure, revised 
edition, 1922. 

R. Brauns. Mineralogie, 5th edition, 1918. 

E.H.BoEKeE. Grundlagen der physikalisch-chemischen Petrographie, 
1915. 2nd edition, by W. Eitel, 1923. 

E. S. Dana. A Text-book of Mineralogy with an extended treatise 
on Crystallography and Physical Mineralogy, 3rd edition, 1922. 

C. DoEtTER. Physikalisch-chemische Mineralogie, 1905. 

B. GossnER. Kristallberechnung und Kristallzeichnung, 1914. 

P. GrotH. Physikalische Kristallographie, 4th edition, 1905. 

P, GrotH. Elemente der physikalischen und chemischen Kristallo- 
graphie, 1921. 


THE CLASSIFICATION OF ATOMS § 189 


H. Hirton. Mathematical Crystallography, 1922. 

F. M. JAEGER. A Treatise on the Principle of Symmetry, 1917. 

F, KLrocKMANN, Lehrbuch der Mineralogie, 7th and 8th editions, 1922. 

ST. KREUTz, Elemente der Theorie der Kristallstruktur, 1915. 

Tu. LiEpiscu, Grundriss der physikalischen Kristallographie, 1896. 

G. Linck. Grundriss der Kristallographie, 4th edition, 1920. 

C, NAUMANN and F. ZrIrRKEL. Elemente der Mineralogie, 15th 
edition, 1907. 

P. NiaGt1. Geometrische Kristallographie des Diskontinuums, Ig19. 

P. Nicei1. Lehrbuch der Mineralogie, 1920. 

F, Rinne. Einfiihrung in die kristallographische Formenlehre und 
Anleitung zu kristallographisch-optischen und réntgenograph- 
ischen Untersuchungen, 4th and 5th editions, 1922. 

A. SCHONFLIES. Kristallsysteme und Kristallstruktur, 1891. 

G. TAMMANN. Kristallisieren und Schmelzen, 1913. 

(GG. TSCHERMAK and F. BecKE. Lehrbuch der Mineralogie, 8th edition, 
1g2t. 

A. E. H. Turron. Crystallography and Practical Measurement 
(2 vols.), 1922. 

W. Voict. Die fundamentalen physikalischen Eigenschaften der 
Kristalle in elementarer Darstellung, 1893. 

W. Votet. Lehrbuch der Kristallphysik, 1910. 

E. A. WULFING. Die 32 kristallographischen Symmetrieklassen und 

ihre einfachen Formen, 2nd edition, 1914. 


[Several of the older books given in the original are 
omitted and some English works have been included 
in the list—TRANSLATOR’S NOTE. ] 


INDEX 


Numbers refer to pages 


A Benzene, 58, 59, 91, 105, 163. 

Benzophenone, 66. 

Absolute zero, 68, 77. Beryl, 14. 

Absorption, optical, 165. Biotite, 152. 

Adsorption, 103, 127, 130. Borazite, 70, 168. 

Adularia, 16. Brittleness, 55. 

Affinity tensors, 82, 94. Bromine, 47, 176. 

Albite, 173. Brucite, 173. 


Albumen, 65, 173, 178. 
Alcohol in crystals, 153. 


Alkalies, 48. C 
Alkali halides, 176. 
Alkaline earths, 47. Cesium, 47. 
Allomerism, 74. — salts, 109. 
Allotropy, 69. Calcium, 47. 
Aluminium, 22. Cale-spar, 7, 25, 31, 57, 90, 91, 99, 118, 
Ammonium oleate, 133. 139, 158, 164, 169, I7I. 
Amorphous bodies, 39. Canals in crystals, 146. 
Anatase, 9o. Cane sugar, 76. 
Andalusite, 132. Carbon, 44, 47, 75. 
Anhydrite, 135, 149. — bisulphide in crystals, 153. 
Anisotropy, 54, 56, 121, 134, 139, 140.|— tetrabromide, 75. 
Anorthite, 173. Carborundum, 72, 117. 
Aragonite, 164, 169. Catalysts, 161. 
Argon, 84, 186. Cell axes, 106. 
Arsenic type, 116. Cells, 106. 
Atom, 40, I41, 181. Cellulose, 65. 
— classification, 180. Centre symmetry, 29. 
— domains, 47, 67, 82, 107, 126, 184. | Chabasite, 153. 
— lattice, 90, 93. Chemical compounds, Ioo. 
— number, 21, 183. —-- formule, 39, 68, 78. 
— rings, 90. — reactions of crystals, 60, 139, 158, 
— sub-types, 185. 170. 
— theory, 5. — — — molecules, 140, 158. 
— types, 182. Chlorine, 47, 186. 
Augite or Pyroxene, 28, 132. Chlorite process, 154. 
Axial sections, 8, 26, 50, 173. Chrysoberyl, 38. 
Chrysolite, 114. 
B Classification of atoms, I8o. 
Cleavage, 7, 27, 55, 91, I6I. 

Barium, 47. Close-packing of spheres, 46, 67. 
— chloride dihydrate, 150, 162. Coarsening of the grain, 131. 

_ — nitrate, 97. Cobaltite, 112. 
Bauerite process, 154. Collective crystallization, 131. 
Benitoite, 31. Colloidal, 45, 130. 


Ig! 


192 CRYSTALS AND MATTER 


Colloidal metals, 46. 
Compounds, chemical, 98. 
Contact, metamorphosis, 131. 
Co-ordination, 80. 

Copper, 22, 118. 

— vitriol, 162. 

Cordierite, 56. 

Crystal classes, 31. 

— nuclei, 211, 128. 

— physiology, 180. 

=~ Structure; §; 

I SVStCiis, 631. 

— undermining, 143. 
Crystalline molecules, 63. 
Crystallisation, 65, 121, 128. 
Crystals, liquid, 63. 
Cuprite, 171. 

Cyanite, 56, 130. 


D 


Debye-Scherrer diagram, 19, 45, 46. 

Decrescence, 7. 

Deformation in physical and chemical 
field, 170, 174, 176. 

— morphotropic, 105, 117. 

Desmine, 151. 

Diamond, 22, 42, 43, 55, 66, 74, 96, 118, 

1S6. 

Dielectric constant, 119. 

Diffraction, 11. 

— equation, I2. 

Digyric axis, 35. 

= SYMIMEClEY,. 27. 

Dolomite, 31. 

Doma, 29. 


E 


Electron shells, 41, 84. 

Electrons, 40, 41, 141, 181. 
Elementary cell, 22, 34, 47, 78, 139. 
Elements, 185. 

Emanium, 86, 

Enantiomorphy, 75. 

Energy quanta, 77. 

Etch figures, 134, 139. 


F 


Fayalite, 114. 

Felspars, 99, 173. 

Fibre diagrams, 17. 

Fine-structural types, 34. 
Fine-structure study, 5. 

Flake diagrams, 17. 

Fluorine, 47. 

Fluor-spar, 22, 79, 118. 

Fore-forms of crystallization, 41, 164. 


Forsterite, 114. 
Friction, internal, 62. 


Fundamental law of crystallography, 
8. 


G 


Garnet, 55, 132. 
Gas, 59, 61. 

Gel, 46. 

Gems, 155. 
Glancing angle, 12. 
Gliadin, 65. 
Globulites, 66. 
Glutaminic acid, 65. 


Gold, 22, 45. 

Graphite, 20, 42, 74, 87, 92, 
155, 

—~) type, 210. 


Graphitic acid, 154. 

Growth, 121, 128, 131, 137; 
— pyramids, 121. 

Gypsum, 32, 57, 134, 149, 159. 
Gyric symmetry, 30. 

Gyroidal symmetry, 30. 


H 


Heemoglobin, 65. 
Halogens, 47. 

Hardness, 55, 62, 154. 
Heat as a catalyst, 161. 
— action, 103, 170. 

— conduction, 56. 
Helium, 84, 140. 
Heterobars, 185. 
Heulandite, 145, 151. 
Hexagonal system, 31, 53. 
Homoomeric, 71. 
Hydrogen, 86, 89, 111, 187. 


Ice, 70, 79, 120. 

Imbibition of albumen, 175. 
Indicators, crystallographic, 163. 
Iodine, 47. 

Ion, 40, 185. 

— lattice, 93. 

Tron, 73, 132: 

— carbide, 78. 

— glance, 122. 

— pyrites, 19, 112, 118. 
Isobars, 185. 
Isodynamostasy, 58, 115. 
Isometric system, 31. 
Isomorphism, 105. 
Isomorphotropy, 105. 


132, 


INDEX 


Isomorphous mixture, 97, 129. 
— stratification, 129. 
Isostasy, 54, 58, 115. 
Isotopes, 97, 186. 

Isotypy, I15. 


Kolnenite, 152. 
Krypton, 84. 


L 


Labradorite, 173. 

Lag points, 77, 100, 150, 156. 

Lattice types, 93, 116. 

Laue, 13. 

ri diagrams, II, 14, 15, 52, 71, 144, 
146, 

— effect, II. 

Lead nitrate, 97. 

Leptyles, 90. 

Leptoblasts, 96. 

Leptology, 5. 

Leptonic axes, 106. 

— volumes, 107. 

Leptons, 5. 

— shape of, 41. 

Leptoscope, 24. 

Leucite, 168. 

Light figures, 134. 

Liquid crystals, 63. 

Liquids, 59, 62. 

Lithium, 47, 184. 

Longulites, 66. 

Loschmidt number, 106. 


M 


Macrostereochemistry, 25. 
Magnesium, 47, 116. 
Magnetic pyrites, 71. 
Margarites, 66. 
Marmorization, 131. 
Mass action, 100, I61. 
Melting process, 178. 
Metabiotite, 152. 
Metabrucite, 149. 
Metachabasite, 153. 
Metaheulandite, 146. 
Metals, 22, 46. 
Metamorphoses, series of, 61. 
Metascolecite, 147. 
Methane, 43, 160. 

Mica, 56, 152. 
Microcline, 38. 

Mimesy, 38. 

Minimum symmetry, 41. 
Mirror symmetry, 29, 33, 40, 140. 
Mixed crystals, 97. 


13 


198 


Mixed crystals, significance of, 98. 
Mixture, physical, 99. 
Modifications, 70, 76, 164. 
Molar axes, 107. 

— volume, 107. 

— weight, 107. 

Molecular lattices, 10, 91, 93. 

— linkage, 94, 103, 130. 

— magnitudes, 48. 

Molecule, 40, 141, 181. 
Molecules, additive, 7. 

— asymmetrical, 27. 

— crystalline, 65. 

— in crystals, 87. 

Monoclinic system, 31. 
Monotopy, 183. 

Morphology, 28. 

Morphotropic constructions, 107. 
Morphotropy, 105. 

Multiple proportions, 79. 


N 


Nitrogen, 47, 142. 
Nuclear charge, 183. 
— sphere, I4I. 


O 


Occlusion, 130. 

Olivine, 95, 113. 

Opal, 44. 

Organic compounds, 24, 90. 
Outer shell, 142. 
Outgrowths, 102. 


Oxygen, 47. 


Parisite, 144. 

Pedion, 29. 

Penecrystals, 63. 
Phenylacridonium sulphate, 164. 
Physical processes, 60, 165, 170, 176. 
Pinacoid, 29. 

Plagioclases, 99, 173. 

Plasticity, 55. 

Point system, Io. 

Polanyi diagram, 18. 
Polymorphism, 69. 

Polytypy, 72. 

Porosity, 126. 

Potassium, 47. 

— bromide, 100, 110, 176. 

— chloride, 100, 102, I10, 176. 
— cyanide, 97, IIo. 

— iodide, 102, I10, 176. 
Pre-chemical processes, 160. 
Primitive bodies, 7. 


194 


Primitive forms, 29. 

Prisma, 29. 

Projection diagrams, 32. 
Pseudoisotropy, 59. 
Pyroelectricity, 130, 147, 148. 
Pyroxene or augite, 28, 132. 


Q 


Quartation, 157. 

Quartz, 27, 31, 36, 44, 57, 70, 72, 73, 
130, 166, 167. 

— type, 116. 


R 


Radical lattices, 94. 

Radicals, go. 

Radium, 108. 

— oxide, 108. 

Rare gases, 86. 

Rationality of axial sections, 8, 26, 50. 

Reactions discontinuous, 160. 

Reconstruction, 153. 

Reflexion in ultra-red, 44, 92. 

— of X-rays, I2, 152. 

— with-translation, plane of, 33. 

Regional metamorphosis, 131. 

Regular polyhedra, 51. 

Resistance to chemical attack, 154. 

Rhombic system, 31, 53. 

Rhythm in crystal structure, 28, 33, 50. 

Rifts in structure, 126. 

Ring structure, go. 

Rock-salt, 22, 26, 47, 55, 82, 83, 97, 
103, 127, 119, 125, 127,120. 

Rotation axes, 28, 33, 50. 

— methods, 16. 

— with reflexion, 28, 30. 

Rubidium, 47. 

— chloride, 108. 

Rutile, 90, 103. 


S 


Salting-out process, 103. 
Sanidine, 147. 

Scolecite, 147. 

Screening action, 126, 155. 
Screw axes, 33. 

Secondary rays, II. 

Silica gel, 46, 153. 

Silver, 22, 46. 

— iodide, 172. 

Sodium, 46, 66. 

— bromide, 97. 

— chloride dihydrate, 162. 
— hydrogen fluoride, 109. 
—- periodate, 31. 

Solution, 133, 137, 177. 


! 
‘ 


CRYSTALS AND MATTER 


Solution forms, 135, 137. 

— process, 136. 

Space lattice, 10, 94. 

Sphenoid, 20. 

Spherical crystals, 66. 

Stability, 58, 60, 70, 84, 115, 129, 136, 
I 


57- 
States of matter, 61. 
Starch, 64. 
Star figure, X-ray, 144, 149. 
Staurolite, 130. 
Step-rule, 72. 
Stereochemical axes, 106. 
Stereochemistry, 5. 
Stereograms, 22. 
Stereophysics, 5. 
Succinic iodimide, 36. 
Sulphur, 47. 
Strontium, 47. 
Structural chemistry, 81. 
— groups, 90. 
— rhythms, 28, 31, 50, 53. 
— rigidity, 141. 
Surfaces of crystals, 79, 125. 
Symmetry actions, 158. 
axes, 29. 
planes, 29. 
point, 29. 
of crystals, 28. 
— leptons, 40. 


T 


Tension processes, 160, 167. 
Tetragonal system, 31, 53. 
Tetragyral symmetry, 31. 
Tetrahedral type, 118. 
Thermal action, 170. 
Topaz, 163. 

Topic axes, 105, 107. 
Topochemical reactions, 143. 
Topotropy, 106. 
Tourmaline, 31, 56, 163. 
Transformation series, 69. 
Trichites, 66. 

Triclinic system, 27, 53. 
Trigonal system, 31, 53. 
Trigyric symmetry, 31. 
Twin formation, 35, 80, 92. 
— gliding, 92. 


U 
Uranium, 43, 85. 

V 
Valency, 80. 
— changes, 87. 


— distribution, 82. 
— tensors, 82, 94. 


INDEX 


WwW Z 


Water in crystals, 144, 149, 162. Zeolites, 144, 155. 


Wave length curves, 56. Zero absolute, 68, 77. 
— valency, 84. 


x Zinc blende, 22, 88, 118. 


— oxide (zincite), 120. 
Xenon, 85. Zone, 12, 173. 


195 


‘ . ate. 


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